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Directions in matrix theory, Auburn 1990, conference report
REPORT
Directions
in Matrix Theory, Auburn 1990, Conference
Report
Frank Uhlig and Tin-Yau Tam Mathematics Department Auburn University Auburn, Alabama 36849-5307 and David Carlson Mathematical Sciences Department San Diego State University San Diego, Calgornia
92182
Submitted by Richard A. Brualdi
1. INTRODUCTION
The
idea for the
conceived There,
a large number
and common however,
fourth
linear
algebra
conference
of researchers
language
that as wonderful
as these
was clearly
conferences
are,
or the direction
as discontinuous
apparent,
they give only a momentary But
will not yield the derivative f
And a direction
as research
over
that our area is experiencing.
of many discrete values off
of f. One has to work harder.
assess for something
was
and the vitality
It became
As one attends such conferences
the years, one might get a feeling for the development in calculus-knowledge
evident.
University
(see LAA, Vol. 121).
from many lands had gathered,
of our research
glimpse of the state of the art in linear algebra. -as
at Auburn
by Frank Uhlig during the 1987 Valencia Conference
developments
is especially difficult to in a vast area such as
linear algebra. So the idea was conceived
of inviting many experts in linear algebra to a conference
and asking them to present their visions of the forces from the past and those leading into the future as regards linear algebra. Frank talked with several of the participants
in
Valencia and wrote many letters from Coimbra in the fall in order to plant the seeds for this conference. “Hilbert’s
The responses were encouraging.
shoes”
would
speaker speak from his/her research:
“What
not fit anyone.
There was a general realization that
But Frank
kept on, suggesting
that every
own standpoint and develop the vision that we need for our
are the important problems?”
LINEAR ALGEBRA AND ITS APPLICATIONS
“Why are they important?’ 162-164:711-797
“Where
(1992)
0 Elsevier Science Publishing Co., Inc., 1992 655 Avenue of the Americas, New York, NY 10010
0024~3795/92/$5.00
did 711
712
FRANK
UHLIG,
TIN-YAU
TAM, AND DAVID
CARLSON
they come from?’
“Where might they lead?” “ What is linear algebra as a discipline capable of solving now or in the near future?” “What lies ahead?” “How are past results pointing us in certain directions?” “ What direction is matrix theory taking?’ Thus the title for our conference. number
In this Special
Issue you will find an unparalleled
of deeply visionary papers on the affairs of linear algebra.
a year after the conference,
it is apparent
that “Hilbert’s
And now, more than
shoes” are not really needed.
We can each wear our own. One important ence.
The
Meetings
new direction
Undergraduate in Louisville
Undergraduate renewed
Algebra
in January
1990,
Conferences
in the teaching
Thompson
came
lectures
conference
aboard
and feature
all active
much needed
Together
research
of 1988.
With him he brought
in TEX and word processing. There
were
33 invited
given, with a total attendance From the beginning to support
of linear
hoped-for camellias
support
in lieu of speakers’
Our financial Auburn The
faltered, support
Uniuersity
We
specifically
Auburn University
to give
in the fall
speakers
for the success
65 contributed
of the
talks were
from 25 countries.
to use the funds that might become
according
our senior
and others
and helped us greatly with his
In addition,
of 135 matricians
speakers
algebra.
Tam joined
speakers.
the
the program of the
Tin-Yau
talents
a
issues.
during
activities,
we had planned
our invited
of 1988
on regional
Most of all we have to thank our group of invited conference.
also with teaching
we tried to balance
his extraordinary
on the
Matrix Theory
to report
surveys of some areas.
expertise
that future
in the summer
areas
at the Joint
in August 1990, represent
will involve themselves
at Johns Hopkins.
at our confer-
Discussion
and Mary Workshop
We expect
as a coorganizer
wanted to offer some minisymposia
Panel
in Williamsburg
of our subject.
at Auburn and elsewhere Carlson
too late to be represented Curriculum
and the William
Linear Algebra Curriculum
interest
Dave
appeared
Linear
to their
speakers
need.
came
And when
through
one
available agency’s
for us by accepting
fees. What a treat it is to work with matricians!
came from four sources:
and its Vice-President
Oak Ridge Associated
of Research (ORAU)
Unioersities
supported
supported
us generously.
us with
unrestricted
funds, a fact that was very helpful in the initial stages when we had to build our logistics base here and had to print and mail out the announcements
etc. in order to even have
the conference. The
National Security Agency
supplied
adequate
funds for our speakers’
expenses
where needed. Finally we are grateful to the Linear Algebra Group of SlAM and to ZLAS, who each helped us out with mailing labels and general And, last but not least, the participants Our sincere
thanks to all that were involved for making this conference
Let us conclude linear algebra and
Linear
Figure
support. who came to attend the show.
our preface
to the conference
over the last 20 years. and Multilinear
Algebra
1 shows graphs of the bookshelf
year intervals,
and with
the number
The journals were
report
Linear Algebra
founded
in 1968
space occupied of volumes
with a factual
possible. assessment
of
and Its Applications
and 1973
respectively.
by each marked on the left in 5
per period
marked
on the
right.
AUBURN
1990 CONFERENCE
ON MATRIX
713
THEORY
50
100 90
Shelfspace in cm
80
40
60 -
30
Number
50 40
-
30
-
of
volumes 20
10
20 10-
1
O-
1968 1972
1982
1 977
0
19t
(a)
30
-
25
-
- 15
10
20 -
Shelfspace in cm
15
-
-
Number of volumes
t
10 5 1 OJ
1973
31977
1982
1987
‘90
(b) FIG.
Cumulative
1.
Linear
and Multilinear
These
graphs
seem
the two journals numbers
totals
by period:
Algebra,
1973-1990.
to suggest
doubles
of participants,
that
about every
These progressions
A doubling
the
This is corroborated
1968-1990;
of papers
in each
and talks given at the Auburn growth pattern
for a constant
a = (In2)/10
of
Conferences
in linear algebra:
. volume of papers(year
(b)
by data on the
2 below.
t) = (1 + tr)
every 10 years is achieved
volume
Appl.,
10 years.
suggest an exponential
volume of papers(year
Algebra
currently
home countries,
from 1970 to 1990, as shown in Figure
(a) Linear
(t = 0.07.
1)).
FRANK
150
UHLIG,
TIN-YAU
TAM, AND DAVID
CARLSON
100I-
25
90I120
80
20
90
60
15
a
lo
ii :: t
50 60
40
r
30 30
20
0
1970
Note that Bob Thompson, an exponential
was obtained
19’80
Auburn Conferences,
in the introduction
19’S;
0
19;o
1970-1990.
to his address for this conference
growth rate for linear algebra with d = 0.04.
by counting
s
&‘--
FIG. 2.
established
::
5
A
__--
10
0
rf IFi
70
the number
of reviews
in the 15Xxx section
also
His rate figure in MathernaticaZ
Reoiews each year from 1940 on. It stands to reason that, given an overall yearly growth of 4%, a much larger growth rate of 7% is obtained Linear
Algebra,
as these were founded
only recently,
The main part of this report contains abstracts
for the two journals
specializing
in
in 1968 and 1973.
the following:
of invited talks, titles of contributed
talks, and synopsis of both.
Of course, we do not repeat those abstracts
or titles of talks that evolved into papers
in this special issue.
2. ABSTRACTS Generalized
OF
INVITED
TALKS’
inverse Invariances
by Jerry K. Baksalary.2 A survey is given of criteria
for the concepts
singular values, etc. to be invariant
with respect
such as range, rank, trace, eigenvalues, to the choice
of a generalized
inverse
‘Only those abstracts are given here that did not result in a paper in this issue. “Department PL-65-069 SF-33101
Zielona Tampere
of Mathematics G&a,
Poland,
10, Finland.
and Statistics,
Tadeusz
and Department
Kotarbiliski
of Mathematics,
Pedagogical University
University, of Tampere,
AUBURN
1990
B-,
they
when
type
CONFERENCE are referred
are relevant
estimability,
ON MATRIX
to in the product
to several
statistical
characterizations
and properties
AB-C.
problems,
of the
of canonical
715
THEORY The invariance
for instance
minimum-dispersion
correlations
in linear
properties
to criteria
linear
of this
for unbiased
unbiased
estimators,
models.
LAP&X-a Portable High-Performance Linear-Algebra Library by Javes Demmel.3 The goal of the LAPACK project library
for efficient
based
on the
eigenvalue including
used
problems,
and
recent
SVD algorithm tion ric
in the library.
First,
and the grading which
thought
positive
contain
definite
small
significantly
singular
improve
using
criterion) and
SVD.
we
show
that
work
with
overview
errors.
Third, more
In fact, even
much
project,
algorithms which
respects
a new bidiagonal
more
accurately
than
differential
we show accurate
as long
using
equa-
that
Jacobi’s
for the symmet-
as the
infinite
is
solving,
of the
solver
a Hamiltonian
is uniformly
library
high-accuracy
we present
and vectors
of roundoff
stopping
a brief on new
Second,
involves
linear-algebra The
for linear-equation
a linear-equation
values
The proof
errors,
After
we discuss
eigenproblem
relative
squares.
we concentrate
propagation
(with a modified
a portable computers.
EISPACK packages
of the problem.
computes
possible. the
and
least
results,
to understand
method
linear
and implement
of high-performance
LINPACK
benchmark
the sparsity
previously
is to design
on a variety
widely
to be included both
use
matrix
precision
entries will
not
on Jacobi.
This talk represents Deift,
J. Dongarra,
L.-C.
Li, A. McKenney,
joint
J. Du
Croz,
E. Anderson,
I. Duff,
D. Sorensen,
M. Arioli,
A. Greenbaum,
C. Tomei,
Z. Bai, J. Barlow,
S. Hammarling,
W.
P.
Kahan,
and K. Veselic.
Structured Linear Algebra Problems in Signal Processing and Control4 by Paul van Dooren. We give a survey processing though
and
the problems
no longer
make
3Department Current address: 94720.
of a number
control, use
where
one wants here
of linear-algebra the
structure
to solve for these
of standard
problems
of the
matrices
linear-algebra
occurring
matrices tools,
involved
are rather since
the
in digital
signal
is crucial. classical, structure
Al-
one can of the
of Mathematics, Courant Institute, 251 Mercer St., New York, NY 10012. Division of Computer Science, University of California, Berkeley, CA
4This paper appeared in Numerical Lirwar Algebra, Digital Signal Processing and Pam&l Algorithms (G. Golub and P. van Dooren, Eds.), NATO ASI Ser. 70, 1991, pp. 361-384. 5Philips Research Laboratory, Ave. Albert Einstein, gium; current address: Coordinated Science Laboratory, Champaign, Urbana, IL 61801.
4, B-1348 Louvain-la-Neuve, BelUniversity of Illinois at Urbana-
716
FRANK
matrices
UHLIG,
has to be taken into account.
how structure
affects the sensitivity
TIN-YAU
We discuss
Structured
matrices
around
for a long time
algebra
several
fields.
dealing
with
matrices,
structure
such
of the matrices
but one is then
from loss of accuracy divergence
of the algorithm
The importance gebra problems recognized occur.
When
them.
We
sensitivity
there
then
are fast algorithms
analyze
of a problem
if the
may have
exploited
in general.
LAPACK:
A Linear-Algebra
by Jeremy
on the
is planned
squares problems, The library
systems
common linear-algebra This library,
packages
squares.
LINPACK
These algorithms.
problems
and how
EISPACK
architectures
talks describe routines,
they
discuss
may affect
structure
the that
could
be
called
LAPACK,
for the analysis
equations,
linear
and least-
problems. a uniform
set of subroutines
solving,
networks,
among machines tools
scheme
not only will ease
of different
for evaluating
and widely used
eigenvalue an important
but they were not designed now becoming
to solve the most
on a wide range of architectures.
via computer
have provided
notes
package
algebraic
also will provide
the naming
and contain
we briefly
Computers
77 subroutines
linear
more portable but
of structured
and control
also look at the effect
computational
FORTRAN
and to run efficiently
for linear-equation
on serial machines,
and vector
proposed
to provide
efficiency,
and
a number
on a matrix
We
The library will be based on the well-known
EISPACK
computing
of
of simultaneous
make codes
and increase
performance.
the proposed
which will be freely accessible
code development,
of structure
linear-al-
and Z. Bai.’
and matrix eigenvalue
is intended
of structured
for these problems,
of an algorithm
to be a collection
of various
suffer
dealing with them is being
Library for High-Performance
outlines
fast algorithms
of the sensitivity
for such a matrix.
Ducroz, Ed Anderson,
This minisymposium
parallel
so-called
the
or in other
which then results in complete
in algorithms
available
stability
for
with exploiting
of the problem,
of digital signal processing
constraint
defined
in
derived
answer.
understanding
in which problem
structure
tures,
of these
and of the error propagation
and indicate
solution
concerned
execution,
from the correct
of a correct
have been
more and more these days. Here we first consider
matrices
which
mainly
Yet several
during their (real-time)
and are encountered
algorithms
in order to improve the complexity
words to speed up the algorithm.
should
constraint.
In linear
application
and show
at hand and how algorithms
have been
various
CARLSON
some of these problems
of the problem
be adapted in order to cope with the structure
TAM, AND DAVID
problems,
LINPACK
and
and linear least
infrastructure to exploit
architeccomputer
for scientific
the profusion
of
available. for the routines,
on the structure
give listings
of the routines
for a few
and choice
of
In addition, a discussion of the aspects of software design is given.
‘Department of Computer Science, University of Tennessee, Knoxville, TN 37996, Oak Ridge National Laboratory, Mathematics Science Section, Oak Ridge, TN 37831.
and
717
AUBURN 1990 CONFERENCE ON MATRIX THEORY A Summary
of Research on Linear Algebra and Matrix Theory in Spain
by Vicente Her&&z7 This talk is devoted working
on topics
to giving a picture
related
paid to the research
with matrix
in the Universidad
of the different
theory
PolitCcnica
Some Recent Results on Singular-Value by Roger Horn.’ A quasilinear representation clearer
understanding
hold between inequality
from which
Special
attention
is
de Valencia.
invariant
norms
Ky Fan domination
for all unitarily
many known
groups in Spain which are algebra.
lnequulities
for unitarily
of the classical
two matrices
and linear
invariant
on matrices
theorem
norms.
and new inequalities
leads to a
on inequalities
that
It also leads to a master
can be extracted
as special
cases. Basic notions essential
of duality play a key role in obtaining
in deriving
an apparently
ucts that obey a fundamental
new characterization
majorization
to treat the ordinary and Hadamard generalization
of both products
inequality.
products
cannot
FFTs and the Sparse-Factorization
our results,
and they are also
of those bilinear
matrix prod-
This characterization
permits
us
in a unified way and shows why a natural
satisfy the basic inequality.
Idea,9
by Charles van Loan.” The FFT Algorithms subscript
literature
tend notations.
sen block-matrix Borrowing framework DFT
is vast, disconnected,
to be
detailed
ideas
from
“sparse factorizations.” and its connection
selected FFT
authors,
algorithms.
with
an array of tricks.
obscure
this situation
I have developed The central
of sparse matrices.
multidimensional through
a well-cho-
The theoretical
vehicle
of this activity
FFT algorithms.
has important
a high-level,
unifying
idea is the factorization
Different
FFTs
correspond
that surface
of the
to different
for doing this is the Kronecker
with the kind of data transpositions
fringe benefit
factorizations
level
notation.
for describing
of vector/parallel
and (to an outsider)
scalar
This talk is about how to correct
matrix into a product
important
at the
in FFT
product work. An
is that our notation facilitates the development
So once again we see that the language
computational
overtones.
‘Dpto. Sistemas Informaticos y Computation, Camino de Vera, s/n, 46071 Valencia, Spain.
of matrix
Notation is everything.
Universidad Politecnica
de Valencia,
‘Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218 with Roy Mathias and Yoshihiro Nakamura).
(jointly
‘This paper gives an overview of the book Matrix Frameworks for the Fast Fourier Transform, SIAM, Philadelphia, 1992. “Department
of Computer Science, Cornell University, Ithaca, NY 14853.
718
FRANK
UHLIG,
TIN-YAU
TAM, AND DAVID
CARLSON
Matrix Complements by Carl Meyer.” The
purpose
is to introduce
complementation
tion is focused on stochastic a partitioned
some variants
in block-partitioned
irreducible
matrices
complementation,
stochastic
concept
special
of Schur
structure.
Atten-
which, in its simplest form, is defined on
matrix
Pll
P=
p12
p
21
i with square diagonal blocks.
of the well-known which possess
The stochastic
p22
1
complement
of Pii in P is defined to be the
matrix
sij = Pii Stochastic as important
+ P,j(
complements stochastic
complementation
z-
for
Pjj) - lPji
i = 1,2 and j = I,2.
have a variety of interesting
interpretations.
are discussed,
Several
theoretical
of the algebraic
and then applications
properties
aspects
as well
of stochastic
to Markov-chain
problems
are
developed. Extensions
of these
ered. The concept
ideas to general
nonnegative
of Perron complementation
well as its applications
Lijwner-Ordering
irreducible
matrices
are consid-
is put forth, and some of its properties
as
are presented.
Monotonicity
and
Convexity
Properties
of
Some
Matrix
Functions by K. Norah-iim.‘2 A survey
is given
some matrix functions the statistical
of LGwner-ordering encountered
literature
set of Hermitian
Combinatorial
monotonicity
in statistics.
only for nonnegative
and convexity
Some of these properties,
definite
matrices,
properties
of
considered
in
are here extended
to the
matrices.
Perron-Frobenius
Theory
by Hans Schneider.13 Combinatorial
spectral
a matrix to its spectral
“Department
theory is the study of the relation
properties.
of Mathematics,
of the graph (or pattern)
In this talk, we mainly consider
(reducible)
of
nonnega-
North Carolina State University, Raleigh, North Carolina,
27695. “Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland; current address: Institut fur Mathematik, Universitgt Augsburg, Memminger Str. 6, D-8900 Augsburg,
Germany.
13Department
of Mathematics,
University
of Wisconsin,
Madison,
WI 53706.
AUBURN
1990 CONFERENCE
tive matrices
ON MATRIX
(or, equivalently,
singular
M-matrices)-as
relate the graph of such a matrix to the structure spectral
radius.
We consider
properties,
such
particular
interest
(combinatorial)
positivity
are known.
possible
relations
between
height
Many conditions
We give a solution
by the title. eigenspace
of the eigenspace
of the Jordan
of the (spectral)
level characteristic.
teristics
part
indicated
of the (generalized)
properties
as the corresponding is the relation
719
THEORY
and its spectral
normal
(Weyr)
We for its
form.
A topic
characteristic
of
to the
for the equality of the two charac-
of a long-standing
problem
to characterize
all
the two characteristics.
Combinatorial Aspects of Multilinear Algebra by Jose Dias Da Silva. l4 We report Other cerning
on some recent
combinatorial
aspects
the permanent
results
on the multilinearity
of multilinear
spectrum
algebra
and the spectrum
On the Equality of the Ordinary
partition
are mentioned, of matrices
Least-Squares
of a character.
namely,
associated
those con-
with graphs.
Estimator and the Best Linear
Unbiased Estimator I5 by George Styan.”
It vector
known
is well
in the general
that the ordinary Gauss-Markov
least-squares
matrix V can be the best linear unbiased the identity
matrix.
dispersion izations
of the OLSE
of these conditions,
Block
We also consider
similarity
It comes
single square
“This
Zyskind.
the situation
of the mean
definite
dispersion
even if V is not a multiple perspective,
the model matrix
is a generalization
matrix
X nor the C.
several simple character-
and a rather
complete
set of
when all or part of X is allowed to vary.
of the usual similarity
of square
matrices
a matrix pair (A, B),
A square,
rather
than a
appeared
in the
A. The first studies on this equivalence
talk is based
just published
We present
of
the development
Problems on Block Similarity
up when we consider
“Department Canada
Neither
along with various examples
“Department of Mathematics, 30 Piso, 1700 Lisboa, Portugal. Finland)
in a historical
to be BLU.
Rao, and (the late) George
Some Outstanding by Ion Zaballa.‘? fields.
(OLSE)
matrix V need be of full rank. The key results are due to T. W. Anderson,
Radhakrishna references.
(BLU) estimator
In this talk we discuss,
of the several conditions
estimator
linear model with nonnegative
Universidade de Lisboa, Rua E. Vasconcelos,
on a joint
paper
(with discussion
of Mathematics
relation
and
with
Simo
and rejoinder) Statistics,
Puntanen
in Amer. McGill
(University Statist.
University,
over
Bloco CL
of Tampere,
43:153-164
(1989).
Montreal,
Quebec,
H3A 2K6.
17Department Vitoria-Gasteiz,
of Mathematics, Spain.
Facultad
de Farmacia,
Universidad
de1 Pais Vasco,
01007
720
FRANK UHLIG, TIN-YAU TAM, AND DAVID CARLSON
control
theory literature
and dealt with linear time invariant
systems of the form
f+) =Ax(t) +h(t). In this context
the equivalence
we say that block-similarity pencils
is usually called feedback
or feedback
equivalence.
In matrix theory
is the strict equivalence
of singular
of the form [sZ, - A, - B].
During relation.
the last years a number
This talk presents
OF
CONTRIBUTED
Extreme
Points of a Set of Positive
by William Fairleigh George
N. Anderson,
Dickinson
have been working
on this equivalence
results as well as some open problems.
TALKS”
Semidefinite
Matrices
Jr., Department
University,
Teaneck,
of Mathematics NJ 07666
and Computer
Science,
(jointly with T. D. Morley and
E. Trapp)
Operators
by LeRoy
Preserving
B. Beasley,
UT 84322-3900 Inversion
of people
some of the obtained
3. LIST
Linear
equivalence
Idempotent Department
Matrices
over Fields
of Mathematics,
Utah State University,
Logan,
(jointly with N. J. Pullman)
of Infinite
Matrices
by Kerry G. Brock,
Department
of Mathematics,
Georgia
Institute
of Technology,
Atlanta, GA 30332-0160 On the Convergence by Jack 36849 The
B.
of the Iterative
Brown,
Proportional
Department
Fitting
Procedure
of Mathematics-FAT,
Auburn
University,
AL
fjointly with P. Chase and A. Pittinger)
pth Roots of a Matrix by Bryan 50011
Elementary
Cain,
Department
Divisors
and Ranked
by Keith L. Chavey, son, WI 53706 Circularity
Iowa
State
University,
Ames,
IA
of Mathematics,
to Matrix Compounds
University
of Wisconsin,
Madi-
(jointly with R. A. Brualdi) Chien,
111, Republic
Study of Complex
Posets with Applications
Department
of the Numerical
by Mao-Ting Taiwan
of Mathematics,
(jointly with T. Laffey)
Range Department
Symmetric
by Dipa Choudhury, 4501 N. Charles
of Mathematics,
Soochow
University,
Taipei,
of China Matrices
Department
Str., Baltimore,
of Mathematics,
Loyola
College
in Maryland,
MD 21210-2699
‘*Only those are listed here that are not represented as papers in this issue.
AUBURN
1990
CONFERENCE
Determinantal
Inequalities
ON MATRIX
for Positive
Cox rg, Department
by B. Ann
721
THEORY
Definite
Matrices
of Mathematics,
Brigham
Young
University,
Provo,
UT 84602 Diagonahzing
the Adaptive
by Jerome Park, Some
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AUBURN
1990 CONFERENCE
More on the Reverse-Order
ON MATRIX
Law
by Hans J. Werner,
Institut
Bonn, Adenauerallee
27-42,
Symmetric
Hankel
Operators:
4.
La Jolla, CA 92093,
SYNOPSES A DIXIE CUP:
consider
We
the
set of all
symmetric
for future
n
Universitlt
(jointly with J. W. Helton)
INTERVALS
x
real
n
convenience.)
OF MATRICES
and G. E. TRAPP23
symmetric
space with inner product
as an
n(n + 1)/2-
(A, B) = tr(AB)/2.
matrices
(The factor
We shall use the following
basis of the real
2 X 2 matrices:
A,=
Note that this is an orthonormal
[
;
_;
1 ,
basis with respect
If A and B are n x n real symmetric
matrix is termed
denote
n
the
set
of
positioe
n
x
symmetric
B - AEPos,.
The set Pos is a cone, i.e.,
X > 0. Thus
A 6 B is the sort of order
and
1 1
A,=
;
:,
to the above inner product.
matrices,
for all vectors x. A real symmetric
relativity:
Research,
Completions and Eigenstructure of Mathematics, University of California,
JR.,21 T. D. MORLEY,22
real inner product
of two is chosen
und Operations
Bonn 1, Germany
VISUALIZINGORDER
by W. N. ANDERSON,
dimensional
fur 6konomie D 5300
Minimal-Norm Department
by Hugo J. Woerdemans’ San Diego,
723
THEORY
we say that A < B if Ax
* x < Bx * x
positioe if A 2 0. We let Pos = Pos,
matrices;
then
A Q B if and only if
Pos + Pos C Pos, and hPos C Pos for all real that one usually
encounters
first in special
if we think of A + Pos as the light cone of future events to A, then A < B if
and only if B is in this cone. To visualize
the cone Pas,,
we simply note that
X = ArA, + AsA, + AsA, =
h
+
x,
h
A3
3
is positive
A,
-
h
1
if and only if Xi + hs > 0, X, - hz > 0, and A; - % - A” 2 0. Putting these
20Current address: Department burg, VA 23185.
of Mathematics,
21Department of Mathematics Teaneck, NJ 07666.
and Computer
22School of Mathematics, morley~cerc.uvu.uMet.edu. 23Department town, WV 26506;
of Computer
Georgia Science
[email protected].
Institute
College of William and Mary, WilliamsScience, of
and Statistics,
Fairleigh
Technology, West
Virginia
Dickinson Atlanta,
University, GA
University,
30332; Morgan-
724
FRANK
conditions
together,
UHLIG,
we see that the cone
TIN-YAU Pas,
TAM, AND DAVID
CARLSON
is simply the geometric
((A,, &> A,) I Al 2 0, AI 2 22 + A?,. If A is a positive symmetric matrix, we define the order inter&,
cone
denoted
{X =
[O,A], by
[O,A] = {X]O
study
generalized (with respect
where
to a suitable
A,, : Y-
e.g., A,, Either
the structure
of [O,A], we need
Y,
the matrix
matrix viewpoint
Ai2:
basis) write A as
YL++
etc.
Y,
viewpoint requires,
of shorted
(If we think of A as a linear where P is the orthogonal
or the linear-operator
however,
bases.)
where dagger denotes
to Y,
operator
operator
the Moore-Penrose the (1,l)
dimensions,
alternative
are needed.
definitions
then we define the parallel We refer the reader
can be expressed
pseudoinverse.
the same as A: then
We also need the concept
viewpoint
entry
then
onto
can be adopted.
Y. The
track of the
as
If A is invertible,
of Y(A)
or
we can
operator,
projection
that some care be taken in keeping
The shorted
partitioned
and A-’
is ([A-‘],,)-l.
is
In infinite
of paraZle2 sum. If A and B are two positive
matrices,
sum A : B as A : B = A(A + B)+B.
to [l, 21 for the basic facts about the shorted
operator
and the
sum.
If VE Pos is a convex set, i.e., an
concept
orthogonal
is simply PA restricted
various orthogonal
parallel
the
Schur complement. If A is a positive matrix, and Y is a subspace,
XV+
extreme point of V if whenever
then Y = Z = X. Intuitively,
(1 - A) VS
@? whenever
0 < X Q 1, then X is
X = hy + (1 - A)Z E $? for some Y and Z in Y,
the extreme
points of V are the corners
theorem
THEOREM
Let A be a positioe n x n real matrix. Then X E [O,A] is an extreme conditions are satisfwd:
1.
is from [3]; see also [7, 91 for related
of V.
The following
results.
point of [O,A] if and only if any of the foU&ing
(a) Then matrix X is the shorted operator of A to some subspace Y. (b) X : (A - X) = 0. (c) X = A1f2PA1/’ for some orthogonal projection P. (d) range(X)
tl range(A - X) = (0).
The above theorem
allows us to give a picture
Let +A denote the linear function
of [O,A]. Suppose that A is invertible.
@A : X - A1/‘XA1/‘. Note that a;’
= +*-I.
It is easy
to see that @A maps [O,I] to [O,A]. Thus the convex structure of [O,I] is the same as the convex
structure
of [O,A]. But the shorts of I are precisely
the orthogonal
projections
AUBURN
1990 CONFERENCE
FIG. 3.
(see [l]); indeed,
ON MATRIX
A polygonal approximation
to the set [O,I] for 2 X 2 matrices.
the fact that the orthogonal
projections
is known independently
of shorted
operators
projections
Ps =
Expanding
I 1 cos 8
sin 8
[COST
p = tI
extreme
We now consider
matrices)
we notice that the rank 1
co?e
sin e cos 8
sin e cos 8
sinse
1
sin@] =
and using trigonometric
A, + (sin20)Aa
we get
+ (cos20)Aa
points of [OJ] are a geometric
the intersection
of the two equal expressions
of two order intervals.
circle. The following result is due
(see [2]), we write A : B : C for either
(A : B) : C or A : (B : C).
24The figures in this paper from the second
identities,
I’
2
to T. Ando [4]. Since the parallel sum is associative
available
points of [O,I]
are of the form
out the matrix elements
Thus the rank-one
are the extreme
(see [8]).
To see this picture 24 (for the 2 x 2 symmetric orthogonal
725
THEORY
author
were produced via E-mail.
by MathematicaTM.
The computer
codes
are
726
FRANK UHLIG, TIN-YAU TAM, AND DAVID CARLSON THEOREM 2.
Let A and B be two positive n x n real matrices, and set V = ‘8”. B
= [O,A] n [O,B]. Then the matrix X E V is an extreme point of V ifand following
conditions
(a) range(A - X) fl range(B (b) (A - X):(B
- X):X
The above theorem
- X) Cl range(X)
either
allows us to explicitly
of V=
therefore
construct
semidefinite
the extreme
matrices.
points in the case
If either A or B is singular,
%“,B = [O,A] fl [O,B] re d uces to the one-dimensional
A < B or B < A, then
[O,A] n [O,B] reduces
that A and B are invertible,
the extreme
= (0).
= 0.
n = 2. Let A and B be 2 x 2 positive then the geometry
only ifany ofthe
are true:
and that neither
to the previous
case.
case. If Assuming
A < B nor B < A, we can classify
points of V by their rank:
Rank 0:
The zero matrix.
Rank 1:
For each angle 0, let Y0 Since ble.
be subspace
spanned by the vector
[cos 0, sin OIT.
YO(A) and Y@(B) are rank one with the same range, they are comparaThe
operators
min{ Y@(A), Y@(B)}
are precisely
the rank-one
extreme
points. Rank 2:
The set of 8’s for which min{ Y@(A), YO(B)} = YO(A) is an interval, For each such 0, let &, be chosen B}.
Then the rank-two
extreme
points are precisely
as 0 ranges over 0 E (@o, 0,).
At the extreme
extreme
in this
points
constructed
[0,, 8,].25
to solve max{ A( YO(A) + h[ A values of 0, i.e.,
manner
Y@(A)] <
Y@(A) + b[ A -
reduce
YO(A)]
B0 and 8,, the
to rank-one
extreme
points. The above theorem construction extreme
generalizes
points generalizes
modifications.
to a construction
The
of the rank-n
points.
We now work out a numerical write (x,
to the n x n case, with suitable
of the rank 2 extreme
y, z) for the matrix
example.
xA,
Let A,, A,, and A, be as before, and let us
+ yA, + zAa. Let A and B be the matrices
A = (2,1,0)
= 2A, + A, =
and B = (2, -
1,0)
= 2A, - A,=
Then [O,A] n [O,B] is, in terms of coordinates, (z - 2)2 > (y -
I
:,
the intersection
;
I
. of the cones
1)2 + z2,
(x-2)2~(y+1)2+z~, x2 2 y2 + 22, x > 0.
25Via the correspondence 0 ++eie, we consider angles as points on the unit circle; thus an interval means an interval of the unit circle.
AUBURN
1990CONFERENCE
FIG. 4.
ON MATRIX
[O,A] rl
This is easily solved. The rank-one
and the same with the y-coordinate points is the piece
“triple
the points
727
[O,B], top transparent.
extreme
points are pieces
negated.
The “upper
of the two ellipses
handle”
of rank-two extreme
of the hyperbola y = 0,
between
THEORY
z = $,
(r
- 2)” - 22 = I
y = 0, z = + f.
The ellipses
and hyperbola
meet
at the
points”
We close with a picture
of
[O,A] fl[O,B].
REFERENCES
1 W. N. Anderson, Jr., Shorted operators, SIAM J. Appl. Math. 20:520-525 (1971). 2 W. N. Anderson and G. E. Trapp, Shorted operators II, SIAM J. Appl. Math. 28:60-71
(1975).
728 3
FRANK N. Anderson
W.
and G.
semidefinite operators,
E.
UHLIC, Trapp,
TIN-YAU The
TAM, AND DAVID
extreme
Linear Algebra A&.
points
106:209-217
4
T. Ando, unpublished communication.
5
G. P. Barker, The lattice of faces of a finite-dimensional 7:71-82
6
S.
L.
Eriksson
and H.
Math.
7
W. L. Green
8
Gert
9
of a set of positive
(1988). cone, Linear Algebra Appl.
(1973).
addition,
Sot.
CARLSON
Monogr.
Extreme
C*-Algebras
theoretic
of Operators
approach
points of order intervals,
and Their Automorphism
14, ISBN 0076-0560,
Shorts
A potential
to parallel
(1986).
and T. D. Morley,
K. Pedersen,
Pekerev,
Leutwiler,
Ann. 274:301-317
Academic,
and
to appear. London
Groups,
Math.
New York, 1979.
Some Extremal
Odessa,
Problems,
1989,
to
appear. 10
A. Shapiro,
Extremal
Appl. 67:7-18 11
S. Wolfram, Redwood
INDUCED
problems
on the set of nonnegative
matrices,
Linear Algebra
(1985). MathematicaTM-A
City, Cahf.,
System for Doing Mathematics,
Addison-Wesley,
1988.
NORM OF THE SCHUR MULTIPLIER OPERATOR
by T. ANDOz6 and K. 0KUBOz7 Let
M,, be the vector
[bij] E M,, denote product
An B =
space of all n square complex
by A0 B their
Schur
matrices.
(or Hadumard)
product,
For A = [aij], B = that is, the entrywise
[ aijbij]. Then each A EM, gives rise to a linear operator
called the Schur multiplier norm of S, with respect
operator,
to a norm 1) -
defined
by S,(X)
S, on M,,
= A0 X (X E M,). The induced
)(on M, is defined by 11S,\( := sup 11 x 11 Q111A0 X (1.
A familiar and useful norm on M, is the spectral norm:
another
where
useful norm is the numerical
radius:
w(A)
= s;p
)I .
‘(,;‘;(,;”
\I and ( * ) . ) denote the Euclidean
,
norm and the standard
inner product,
26Division of Applied Mathematics, Research Institute of Applied Electricity, Hokkaido University, Sapporo 060, Japan. 27Mathematics Laboratory, Sapporo College, Hokkaido University of Education, Sapporo 002, Japan.
AUBURN 1990 CONFERENCE
729
ON MATRIX THEORY
respectively. We denote the induced norms of SA with respect to the spectral norm and the numerical radius by 1)S,ll_ and 11S,ll w, respectively. It is mentioned in [2] that Haagerup succeeded in determining IIS, II_ in the following form. For A = [aij] EM,,
HAACERUP’S THEOREM.
the following
assertbns
are mutually
equivalent : A = B*C such that
(ii) A admits a factorization
B*BoI&Z
C*CoZ
and
where Z is the identity (or unit) matrix. (iii) There are vectors 2, iji E G” (i = 1,2, . , . , n) such that 1,2, . . . , n) and aij
=
(iv) There are 0 Q R,, R, E M,,
(Zjliji)
(i,j
= 1,&n).
such that
R,oZ
and
In this note we are going to give the characterizations derive Haagerop’s theorem for it as a consequence. THEOREM.
(9, (ii),
II?ziI(, I( GilI Q 1 (i =
R,oZ,
of the norm I(S,II w, and to
For A = [ aij] E M,, the following assertions are mutually equivalent:
II%ll, G 1. A admits a fwtorieation
A =’ B*WB such that B*B-Z&Z
(iii)w There
are vectors
II$1) Q 1 (i = 1,2,.
?ieO”
(i = 1,2,.
. . , n) and a matrix
. . , n), W*W Q I, and aU = (WGjlZi)
(iv), There is 0 < REM,
W*WQZ.
and
(i = 1,2,.
..,n).
such that
[
: 1 A R
20
and
RoZgZ.
WE M,, such
that
730
FRANK
UHLIC,
We can derive from the theorem Schur multiplier (I)
llS,ll,
TIN-YAU
the following
TAM, AND DAVID
properties
CARLSON
of the induced
norms of
operators: C IIS,ll,
Q 2llS,ll0,
(AeM,).
(2) llS,ll, Q II AlI, (AEM,). (3) I]S,]] oD= I(S,]] w if A is Hermitian. (4)
]IS,I] m = ]]S,]] w = 1 if A is unitary.
(5)
llS,ll, G llSI~~+~~*~ llw (AEWJ
(6)
](S,]],.(]SIA~)]u;if
Aisnormal.
The details will he published
in [l].
REFERENCES 1
T. Ando and K. Okubo, Induced
2
Appl., to appear. V. I. Paulsen, Completely Bounded Maps and Dilation, Math.
146, Longman,
Essex,
norms of Schur multiplier
U.K.,
operator, Pitman
Linear Algebra
Research
Notes in
1986.
AN APPLICATION OF VALUATION THEORY TO THE CONSTRUCTION OF RECTANGULAR MATRIX FUNCTIONS by
1.
JOSEPH A. BALL and MAREK RAKOWSK12’ Zntroduction The problem
functions
we address
with a given
complex
plane C,2
B
formed
m
X n
by functions
rational
m x n rational
matrix
and
function
WmXn(u)
WE gnx”,
analytic
functions.
do there
with m
x
exist rational
subset
matrix
c of the extended
exist, how to find one? and by B(u)
the subring
will denote the g-vector on u. Bnx” B? mXn (u) will denote the %?(o)-module
which are analytic
algebraically find
When
over a proper
the field of scalar rational functions
matrix functions
can be characterized Bmx”
structure
If such functions
We will denote by 9 of
here is as follows.
zero-pole
Nw, fiwe
over
BmX”(u),
of of
on 0. The zero and pole structure
in terms of coprime
n matrices
space space
B
factorizations. and
g(u),
Namely, identify
respectively.
DWs kJfflx”(u), and fiw~
Given a gmxm(u)
such that (i) det Dw # 0 and ew,
NW are right coprime
(ii) det D, # 0 and D,, NW are left coprime (iii) W = N,D&’ = fi&‘6w. In this notation, functions
[over g(u)], [over g(u)],
W, HE 92 lnxn have the same left zero structure if NH = NwQ W and H have the same right pole structure
for some Q such that Q, Q- ’ E %! “x”(u).
28Department of Mathematics, Virginia Tech, Blacksburg, VA 24061.
AUBURN
1990 CONFERENCE
if fin = Rfiw
for some
THEORY
731
R such that R, R- ’ E .%’“.“(u)
(see [2] for the regular case,
that is, when the functions vanish
ON MATRIX
involved
are square
and
determinants
which
do not
identically).
A concept null-pole
which
refines
subspace
the notion
W over
of
WS? nx1( a). For the regular
u,
case,
l] (see [2] for a comprehensive matrix functions have been Clearly,
functions
of zero
i.e.
null-pole
in S? mx”
subspaces
the same
subspaces For matrix
over some
functions
methods
right
pole
have played
u is that
9?‘nx1,
of a
yO(W) =
a key role in [7, 8, 3, rational
over u have the
over u. Indeed, if W, HE 9 mx”, then Q such that
left zero
Y,(W)
=
(see Proposition
Q, Q- ’ E W mxn(u)
and HE W mx”ff, then W = HQ for some
structure
S’rm pl e examples over
u may
have
show
that
different
functions null-pole
u. time
the
analytic
has been
for finding
been
and
over of
Null-pole subspaces of nonregular
Q E S? “*x”“( u) if and only if J$( W) C Y,(H). with
structure
with the same null-pole subspace
if WE .%’n’x”u
generally,
and pole
W(u)-submodule
in [6].
investigated
Yo( H) if and only if W = HQ for some 1.1 in [4]). More
the
exposition).
same right pole and left zero structure
have
have
a regular
available.
description
known
One
rational
subspaces
matrix
function
with
there). a given
of regular
rational
Also, constructive null-pole
subspace
to the general case is via embedding in the regular
approach
case. A tool for such embedding
of null-pole
(see [2] and the references
is provided by valuation theory (see [9], [12], and, for
the general theory [ll]).
2.
Orthogonality Let
from
in 2 n
h be a point
of the
where
r) is the unique
integer
r(z)
with
F analytic
complex
stronger
triangle
such
plane.
We define a function
( * 1z=h
=
that
1
(Z-X)“?(z)
if
XeC,
z-V?(z)
if
X=03
at h. The function
and nonzero
1n ) z=h < 1 for every
integer
n, the valuation
) . ) ==Ais a real valuation of W . Since ) * 1z=x is non-Archimedean and the
inequality
I r1 holds
extended
S? into the set of real numbers by putting
for all rr, rs E 9.
+ r2
I z=x Q m=4 I t-1 I z=h, I f-2 I z=hl
732
FRANK UHLIG,
TIN-YAU TAM, AND DAVID CARLSON
Let n be a positive integer and let he C,. the product
We define a function
)] .
1)z=x on W”,
of n copies of ~8, by putting
ll(rIpf-2,.
..,
rn)Ilz=x= m={lrll.=~, lr21z=h....,ImldJ
In this way 9? ” becomes a normed vector space over the real valued field ( 1, Following
the definition of orthogonality
shall say that two subspaces
in a non-Archimedean
A and Q of W * are orthogonal
I * ( ==,,_).
normed
at XE C,
space, we
if
IIx + Y II==h= m={ IIr IIdI II Y IIz=h} for each
r E A, y E Q. We shall say that A and Q are orthogonal
orthogonal
at every point of u. We shall say that vectors
at
(respectively,
XE C,
on a subset
{xi:i#j)areorthogonalat
X(on
o of C,)
u)forj=
W” and AEC,,
Ifhisasubspaceof
on u C C,
if they are
rr, ~2,. . . , rk are orthogonal
if the spans over
W of { xj}
and
1,2,...,k. we will denote by A(A) the set of values at X
of those functions in A which are analytic at A. Plainly, A(A) is a subspace of C”. The space A(A) can be characterized coefficients
equivalently
as the linear span over C of the leading
in the Laurent expansions at A of the functions in A. Using this notation, we
can characterize
orthogonality
in 8 ” as follows (see Proposition
2.3 in [6]).
Let A and Q be two subspaces of W “, and let A E C,.
PROPOSITION2.1.
Then A
and R are orthogonal at A if and only if A(A) fl R(A) = (0). It follows from the definition of orthogonality at a single point AEC, subspaces
of
9”.
necessarily
that two subspaces of W ” orthogonal
have the trivial intersection.
We say that the subspace
Q is an orthogonal
subspace A in (C, a) if A and n are orthogonal the next proposition,
see Proposition
PROPOSITION2.2. proper
subset
u of C,.
Let A, 0, and C be of the
complement
on o and A + Q = X. For the proof of
2.5 in [5].
Let A and Q be subspaces Then Q has an extenskm
of 92 n which are orthogonal
on a
complement
of A in
c C C. Also, Wax”
will
to an orthogonal
(W”, 0). 3.
Analytic Description of Pole and Zero Structure For notational convenience, we will assume hereafter
denote m x n rational matrix functions analytic in C-1
u and vanishing at infinity. By
partial fraction expansions, we have an exact sequence
where all vector spaces are over C. We will use the symbol I’,= for the projection (3.1) for arbitrary positive integers
m and n.
as in
AUBURN
1990 CONFERENCE
If WEWmxn, C-linear
space
ON MATRIX
we define
P,c( yO( W)).
the
733
THEORY
right pole structure
of W over
8,
For th e p roof of the next proposition,
o to be the
see Proposition
4.1
in [6]. PROPOSITION3.1.
8,
= {C,(z The
Plainly,
pair (C,,
A,) in Proposition
denote
the left annihilator
is any matrix polynomial Let
3.1
is called
a right pole pair
such that
for W over
(I.
C u.
u( A,)
Let W”
There exists an obseruable pair of matrices (C,, A,) : x a constant vector}.
- A,)-‘x
of W in g1 xm. A
Zdt kernel polynomial of W
whose rows form a minimal polynomial
XE u be a zero
of W. Choose
(a lxm,X). One can show (see Proposition of matrices (A, Bx) such that
P,c({~En,:4WcWlx”(c)})
an orthogonal
basis (see [9]) for W”.
complement
Ax of W”l
4.3 in [S]) that there exists a controllable
= (~(.a
- A,)-‘Bx:
in pair
z isaconstantvector}.
If x,, &, . . . , X, are the zeros of W in u, the pair
diag(Ax,,A&
,...,
Aht),[B$
BE
a..
B<’ 1)
is called a lej? null pair for W over u. PROPOSITION3.2. One canjnd such thatPO~({q5~A:~W~W1Xn(u)}) For the proof of Proposition Proposition
an orthogonal complement A of W” in ( glx”‘, = {x(z -A,)-‘B,: raconstantuector}. 3.2,
see Lemma
3.16
3.2 a subspace associated with the pair (At,
If (C,,
A,),
(Al.
a)
in [5]. We will call A as in
Br).
Br), and P, are as above, there exists (see Theorems I’ such that rA, - ATr = BrC, and
2.7, 3.1, and
3.3 in [S]) a unique matrix
%(W)
=kerP,n
C,(z-A,)-‘x+h(z):+~C”‘~~,h~W~~~(u),
and c
Res,,,O
(z - A,)-‘B,h(
z) = TX
Z&l A triple ((C,, A,), (AC, Br), r) is called a I& u-spectral triple of W.
4.
Construction A construction
triple
of a Function
is given in [lo].
function Theorem
with prescribed 6.2 in [6].
with the Prescribed
of a regular rational matrix function Below,
we utilize
left u-spectral
Null-Pole
Subspace
with a prescribed
it to construct
a rectangular
triple and kernel polynomial.
left u-spectral rational
matrix
For the proof, see
734
FRANK THEOREM 4.1.
given. left
Let
matrices
Then there exists a rational a-spectral
conditions
0)
triple
and
P,
UHLIG, C,,
A,,
TIN-YAU Ay, B,, r
matrirfunction
TAM, AND DAVID and
with
polynomial
as a left kernel
a matrix
7, = ((C,,
polynomial
A,),
g and
CARLSON P,
be
( Ay, Bc), I’) as a
only
if the following
hold:
the pair
(C,,
A,)
(ii) t& pair (At.
is obseruable
and u( A,)
C u,
and u( Ar) C u,
Br) is controllable
at infinity,
(iii) P, has no zeros in C, and its rows are orthogonal
(iv) the rational matrix function PK( z)C,( z - A,)- ’ is analytic on C, h,} and lk is the identity matrix with the same number {A,,&,..., (v) $a(A&= rows as P,, the pair diag( A(, X,1,,
. . . , A&),
[ B;
P(A1)’
...
PN’]
of
T,
is controllable, (vi) I’A,
- A$
= B< C,.
If rS and P, satisfy conditions 7, as a left u-spectral
triple
(i)-(vi)
in Theorem
and P, as a left kernel
4.1, a function polynomial
WE 9?“x”
with
can be constructed
as
follows.
step 1.
Find a regular Choose
Step 2.
rational
matrix function
a Smith-McMillan
Let Y be the largest largest geometric of the geometric
factorization
geometric
multiplicity multiplicity
H with 7, as a left u-spectral
multiplicity
of a pole of H in u, let /J be the
of a zero of H in u, and let q be the largest sum of a pole and the geometric
of H at any single point of u. Let di denote
multiplicity
9ij
manic polyno-
has a zero at a point XE u of order k then
zero at h of order k. Define
=
an m x r) matrix polynomial
1
if
‘i=jQv-p,
Pi
if
v-r
if
i=j-q+m>m-CL,
1
i 0
of a zero
the ith diagonal entry of D. For
v, let pi be the minimal-degree
i=v--+l,q-p+2,..., mial such that if d,_s+i
triple.
EDF of H, and put W, = ED.
pidi has a
Q = [qij] by
otherwise,
and put W, = W,Q. Step 3.
Find a subspace
E associated
with the pair (A,,
W, onto Ker P along the subspace Step 4.
Find an orthogonal U u( AI)),
complement
and a basis
{ul, 02,.
of W m xl
Bt). Project
annihilated
every column of
by E to get W,.
A of the column span of W, in (Ker P, u( A,) . . , ul} for A such that the function [ul up
. . * ul] has neither zeros nor poles on u( A,) U u( A(). Put W, = [Ws 01 ~2 . . . 011. Step 5.
Multiply
W, on the right by a regular
poles or zeros on u( A,)
poles nor zeros in u \ (u( A,) Particular
rational
matrix
U u( Ar), so that the resulting
steps of this construction
function
function
Q, without
W has neither
U a( A()). are illustrated
in [5] with specific
examples.
AUBURN
1990 CONFERENCE
ON MATRIX
735
THEORY
REFERENCES 1
J. A. Ball,
N. Cohen,
improper
rational
and A. C. M. Ran, Inverse
matrix functions,
Matrix Functions (I. Gohberg, J. A. Ball, I. Gohberg,
3
J. A. Ball, I. Gohberg,
and L. Rodman,
problems
matrix functions,
Sot., 4
and L. Rodman,
Boston,
for rational
zero-pole
Two-sided
OT (I. Gohberg,
functions,
submitted
McMillan structure
Birkhbser,
Null-pole
subspaces
G. D. Forney, I. Gohberg
Jr.,
matrix functions
to appear.
of nonregular
rational
matrix
rational
matrix
problems
for rational
matrix
problems
for rational
matrix
of nonregular
spectral
inverse
spectral
Zntegral Equations Operator Theuy 10:349-415
multivariable functions
rational
(1987).
Linear Algebra Appl. 86:237-282
functions,
10
Math.
for publication.
J. A. Ball and A. C. M. Ran, Local
9
interpolation
to appear.
J. A. Ball and A. C. M. Ran, Global inverse functions,
degree
Linear Algebra Appl.,
Zero-pole
Ed.),
J. A. Ball and M. Rakowski,
8
Lagrange-Sylvester
in Proc. Symp. Pure Math., Amer.
Minimal
structure,
J. A. Ball and M. Rakowski, functions,
7
1988, pp. 123-175.
Znterpolatixm of Rational Matrix Functions,
to appear.
with prescribed
6
for regular
1990.
J. A. Ball and M. Rakowski,
5
problems
Ed.), OT 33, BirkhHuser, Boston,
2
OT 45, Birkhguser,
spectral
in Topics in Znterpolatiun Theory of Rational
Minimal
linear systems,
bases of rational
spaces,
SZAMJ. Control 13(3):493-520
and M. A. Kaashoek,
and minimal
vector
(1987).
divisibility,
An inverse
spectral
with applications
to
(May 1975).
problem
for rational
matrix
Zntegral Equations Operator Theory 10:437-465
(1987). 11
4. F. Monna,
12
G. Verghese
Analyse Non-archimdienne, and T. Kailath,
Rational
Springer-Verlag,
Matrix Structure,
New York, 1970.
in Proceedings of the 18th
ZEEE Conference on Decision and Control, Fort Lauderdale, FL,
Vol.
2, Wiley, New
York, 1970, pp. 1008-1012.
CLASSES OF STABLE MATRICES by ABRAHAM
BERMAN2’a
The inertia, i+(A)
3o and DAFNA
in A, of a square
is the number
of pure imaginary
matrix
of eigenvalues
eigenvalues
SHASHA2’
A is a triple
(i+(A),
io( A), i_(A)),
where
of A in the right open half plane, io( A) the number
of A, and i_(A)
the number
of eigenvalues
in the left
open half plane. A matrix
A E R”* ”
is stable for every positive
diagonal matrix (a diagonal matrix whose diagonal entries
“Department
of Mathematics,
IS (positive) stable if i+(A) = n. A is D-stable if AD
Technion-Israel
Institute
of Technology,
Israel. 30Research
Supported
by the M. and M. Bank Research
Fund.
are
Haifa 32000,
736
FRANK
positive) matrix
D.
A is (Lyapwwv)
D such that
diagonally
diagonally
are D-stable.
in differential
5, 4, 8, 91. A real matrix
study these
preserving
are D-stable.
[semildefinite.
Stable,
equations,
TAM, AND DAVID
if there
@ni]stable
A is inertia-preseruing
in AD = in D. We matrices
TIN-YAU
AD + DAT is positive
stable matrices
arise in problems
UHLIG,
D-stable,
ecology,
chemistry,
is not true, as shown by the following
EXAMPLE 1.
Let
A=[;
Here
A is D-stable
[6] but not inertia
[2], that
diagonal
because,
e.g. [2,
matrix
clearly,
D,
inertia
example.
-5;].
preserving,
since for D = diag{ - 1,3, - 1) the
AD is also stable.
A subclass
of the
[l]: matrices
matrices
D-stable
matrices
A such that
only if D is positive. Manus
1
diagonal
e.g.
and economics,
invertible
This is of interest
The converse
matrix
exists a positive It is known,
and diagonally stable matrices
if for every
matrices.
CARLSON
is the
class
AD is stable,
where
Again, it is clear that inertia
of
Arrow-McManus
D-stable
D is a diagonal matrix,
preserving
matrices
if and
are Arrow-Mc-
D-stable.
Example D-stable.
1 is also an example
Observe
A real matrix preserving
D-stable.
D-stable A is
invertible)
and diagonally
strongly
matrix
inertia
semistable,
preserving
submatrices
that
real
but not strongly inertia preserving.
diagonal
A is strongly
are inertia preserving.
The matrix
is inertia preserving
but not inertia preserving.
if for every
D, in AD = in D. Observe
if and only if all its principal
EXAMPLE 3.
[7]. In fact, we show
Let
A is Arrow-McManus
necessarily
matrix which is not Arrow-McManus
semistable
Let A be a diagonally semistable matrix. Then A is D-stable if and only
THEOREM 1.
if it is Arrow-McManus EXAMPLE 2.
of a D-stable
that it is not diagonally
(not
inertia
AUBURN 1990 CONFERENCE
737
ON MATRIX THEORY
An important class of inertia preserving (and even strongly matrices is the class of diagonally stable matrices.
inertia
preserving)
A diagonally stable matrix is strongly inertia preserving.
THEOREMS.
Clearly, if A is inertia preserving, then io( A) = 0. The following theorem terizes the real square matrices A such that io( A) = 0. Recall [7] that B,(A) denotes the cone B,,(A)
(a) (b) (c) (d)
= (BEPSD[(BA)~~=O,~=I,...,~)
The folhving
THEOREMS.
charac-
properties
of A E R”, n are equivalent:
io( A) = 0.
BEPSD, BA+ATB=O * B=O. BcPSD,BA+ATB=0,rankB<2 t+ B=O. B EB,,( A), BA + ATB = 0, o B = 0. (e) BEB,,(A), BA+ATB=0,rankB<2 o B=O. For matrices which have no pure imaginary eigenvalues THEOREM
4.
one has
Suppose io( A) = 0. Then
(a) in A = in AD for every positioe diagonal matrix D if and only if
(b) io( AD) = 0 fw every positive diagonal matrix D. Given that a matrix A is stable, we obtain D-stability as a corollary of Theorems 3 and 4. Let A be a real stable matrix.
COROLLARYl.
every B E Bo( A) of rank
4.
a simple
characterization
of
Then A is D-stable ifand only iffor
< 2 and for every positive diagonal matrix D, BAD+DATB=O
EXAMPLE
here
o
B=O.
The matrix
is stable. To show that it is D-stable we observe that B E B,( A) if and only if it is of the form a 0 ob’
B=
-4c
0
-4c 0, C
a, b, c > 0, I
a > 16~.
738
FRANK
UHLIG,
TIN-YAU
TAM, AND DAVID
CARLSON
In this case
4c-
a
16c
a0 0
and if BAD is skew-symmetric
for an invertible
are equal to zero, so BA = 0, which implies Since
inertia
matrices,
preserving
matrices
it is natural to ask whether
preserving.
Note that Example
two classes coincide
THEOREM
are
matrix
D-stable
and include
diagonally
this question
acyclic
c, a, and b
the diagonally
semistable
matrices
in the negative.
stable
are inertia
However,
the
matrices.
Let A be an acyclic irreducible
5.
D, then necessarily
that B = 0.
D-stable
2 answers
for irreducible
I I
matrix.
Then A is D-stable if and only
ifA is inertia presweruing. If the irreducible
2.
COROLLARY preserving
if and only if it
The following property
THEOREM
components
of A are acyclic,
of diagonally
semistable
matrices
is of great importance.
Let A E R”,” be a diagonally semistable matrix,
6.
invertible diagonal matrix.
then A is inertia
is D-stable.
and let F be an
The following are equivalent:
(a) in AF = in F (b) io( AF) = 0. (c) BAF + FATB # 0 for every nonzero B E B,( A) s.t. rank B < 2. (d) BAF + FATB # 0 for every nonzero B E B,( A). Observe general,
that
the
implication
as shown by Example
COROLLARY
(b),(c),(d)
i* (a) in Theorem
Let A be a diagonally semistable
3.
6 does
not hold
in
1.
matrix.
The following are equiva-
lent: (a) A is inertia preserving. (b) io( AF) = 0 for every real invertible diagonal matrix F. (c)
BAF + FATB # 0 for
every
nonzero
BE Bo( A) such
that rank B Q 2, and fw
every real diagonal invertible matrix F. (d) BAF + FATB f 0 fat every nonzero B E B,( A), and for every real diagonal invertible matrix F. We conclude
QUESTIONS.
by asking the following
questions:
Is every strongly inertia preserving
matrix diagonally
stable?
AUBURN 1990 CONFERENCE QUESTION 2.
Is every irreducible
The proofs and additional
739
ON MATRIX THEORY inertia preserving
matrix diagonally semistable?
examples will appear in [3].
We wish to thank Professor Daniel Hershkowitz for many suggestions which improved the paper.
REFERENCES K. J. Arrow and M. McManus, A note on dynamic stability, Econometrica 26 (1958). G. P. Barker, A. Berman, and R. J. Plemmons, Positive diagonal solutions to the Lyapunov equations, Linear and Multilinear Algebra 5:249-256 (1978). A. Berman and D. Shasha, Inertia preserving matrices, SZAM /. of Matrix Anal. Appl., to appear. B. L. Clarke, D-stability and chemical reaction networks, presented at the Combinatorial Matrix Analysis Conference, Victoria, 1987. G. W. Cross, Three types of matrix stability, Linear Algebra Appl. 20:253-263 (1978). D. J. HartRel, Concerning the interior of the D-stable matrices, Linear Algebra Appl. 30:201-207 (1980). D. Hershkowitz and D. Shasha, Cones of real positive semidefinite matrices associated with matrix stability, Linear and Mu&linear Algebra 23:165-181 (1988). J. F. B. M. Kraaijevanger, A characterization of Lyapunov diagonal stability using Hadamard products, Linear Algebra Appl., submitted for publication. J. F. B. M. Kraaijevanger and J. Schneid, On the unique solvability of the Runge-Kutta equations, Numer. Math., submitted for publication.
CONJUGATE-SUBSPACE DECOMPOSITION AND ITS APPLICATION SOLVING LINEAR SYSTEMS WITH MANY RIGHT-HAND SIDES by MEI-QIN
IN
CHEN31
The inspiration for the conjugate subspace decomposition (CSD) has come from the updated conjugate subspace method for solving unconstrained minimization problems whose objective functions are twice differentiable, as first introduced in [5]. We assume throughout this paper that A is an n x n symmetric positive definite matrix. The CSD can be described as follows [3].
31Department of Mathematics/Computer
Science,
The Citadel, Charleston,
SC 29409.
740
FRANK THEOREM 1.
UHLIG,
TIN-YAU
Let R” be a conjugate sum of T,,
TAM, AND DAVID
CARLSON
. . , T,,, with respect to A, that is,
(1) R” = T, + *.. +T,; (2)
rfArj
= 0 for xi E Ti, rj E Tj, and i # j.
+ * * . +I(~)
Then x* = 8) XE ‘1;:. The
following
equations
is an algorithm
based
on the CSD
for solving
systems
of linear
Ax = 6. (CSD).
ALGORITHM matrices
solves Ax = b, x E R”, where each xci) E Ti solves Ax = b,
Let R” be the conjugate
sum of T, . . . T,, and Zi be the basis T,, respectively. Let nj = rank Ti, where n, + .- . + n, = n. x* of Ax = b can be computed as follows:
of the subspaces
Then the solution (1) Compute
bj = Zfb, i = 1,. . . , m.
(2) Solve Z:AZ, fci) = bi for ??ci)E R”i, i = 1, . . . , m. I ,..., m. (3) Evaluate x (i)=Z,&i),i= (4) Set r* = x(l) +
*. * +x(@.
The
allows
CSD
strategy
ni-dimensional problems.
subproblems
Some questions,
scale problems
however,
ment in efficiency?
subspaces
a class of problems
x(t)
numbers. when
need to be answered:
The
the
b(t)
(CC)
(BCG)
multiprocessor
In general,
time; for example,
the vector
this case, the BCG algorithms
algorithms
sequential
in
definite, of
P)
the right-hand-side
vector
t, and S is a set of discrete
is suitable
for solving
and without
an explicit
[7] are adaptable
for all
b
real
(P), especially form.
If the
t in S, then the
and work efficiently
on a
the b(t) may not be available at the same
are no longer adaptable.
If the vectors
then the Lanczos-Galerkin
r(tj) for j < i. In b(t) do not change
projection
error bound of the approximation
If this is not the case, then computing
such problems.
the solution
at step (2) to find each solution
It has a high parallelism
of the CSD,
of the CG.
t E S,
procedure
of the solution is
of (P) is completely
t. The algorithm combining the CSD strategy with the CG algorithm, that
is, applying the CG algorithm because
subspaces?
b(tj) may depend on previous solutions
in [9], and its theoretical
given in [lo].
(1) For what type of large
simultaneously
however,
too much from one solution to another, is suggested
method sparse,
are available
block-conjugate-gradient machine.
positive
functions
A in (P) is large,
vectors
m
with less cost and with great improve-
for each
= b(t)
conjugate-gradient
right-hand-side
by solving
for solving large scale
of the form
that are real-valued
matrix
problem
(2) For a given linear system of equations,
be chosen
E R”, A E R “xn is symmetric
has components
n-dimensional
(3) What is the cost of forming these conjugate
Ax(t) where
an
and has great potential
should the CSD be adapted?
how should the conjugate Consider
us to solve
in parallel
and it does not require
in computation the explicit
x-ci), is suitable for solving for solving
(P) for each
form of the matrix
A, because
t
AUBURN
1990 CONFERENCE
ON MATRIX
Notice that the basis matrices set S, the cost of forming So it is important efficiently. the
The performance
spectrum
unfavorable
of
distributions methods matrix
A in the
spectral
well-separated,
Zi need to be formed only once. With many t in the
Ti can be compensated
to choose
of the CG algorithm presence
is very sensitive
of roundoff
errors
In practice,
can be found, for instance, parameters
[l,
the spectrum
when the relaxed
w close
of solving (P).
solves each subproblem to the distribution
6, 8, 9, 11,
for the CC is when the spectrum
eigenvalues.
with varied
by the overall efficiency
Ti’s such that the CG algorithm
distribution
large
741
THEORY
has a few distinct,
of A in (P) with such
(block) incomplete
to 1 are adapted
of
121. One
Cholesky
as preconditioners
to the
A which results from discretizing
-Au=ffor(r,y)EOand
u=Ofor(x,y)~afi,
by linear
finite
examples
with different
element
approximations choices
where
over a uniform
the relation
between
following are two theorems projections approximate
on the
eigenspace
subspaces
the spectrum
triangulation.
Some
one
of the
for a given A, it is necessary
of A and its projections
[3] which describe
Ti’s when
isosceles
x (0,l)
of w are given in [2].
In order to form a set of proper conjugate investigate
0 = (0,l)
the relation subspaces
of the spectrum
contains
to
on the Ti’s. The of A and its
an eigenspace
or an
of A.
THEOREM 2. Let (X,,qJ,i = l,...,m, be the eigenvalue-eigenvector pairs of A, and let Q = [ql. . . . , q,,J. If span Q E Ti f or some i, then span Q I ?; for j + i. lf Zj’Zj = I,,, for each j, and (hi, qi) are extreme eigenpairs of A, then X,, . , . , A,,,are not in the spectrum of A on Tj for j f i.If in addition ZfZj = 0, then the former statement is aLso true for intermediate eigenpairs.
. . ,4,1, w/m-e 1 C ml3 m2
THEOREM 3. Let iT, = [&, . . . , g,,,], & = [q_,,+l,. Q n, and let pk = i$ A& /&& for each k be such that
I pk
-
‘k
I = “$
I Pk
-
$1,
k=
l,...,m,,
1~
k=
and I &t-k
If span[&,
-
h-k
I =
jF;yk
I /hi-k
-
h-k
l,...m,.
G2] E Ti for some i, then fm j # i,
[
k=.,$,,,,,
(:)‘1’:.
742
FRANK
Furthermore, in the inter&
UHLIG,
TIN-YAU
TAM, AND DAVID
CARLSON
if ZiZj = l,,, fm each j, then the spectrum of A on q fw j # i is contained [a, b], where
a= &,+1 -
k$l (xm,+1 - Ak)Ck(Ek)>
where the functions ck are such that
ck(o) = 0
Theorem
2 and Theorem
the CC algorithm, of the conjugate
to (i) choose a subspace
of A whose corresponding proper algorithm
eigenvalues
Ax(t)
ri are normalized = b(t), for some
(1) The vectors -
appear in the spectrum
of A on the
conjugate
vectors
we need
eigenspace
to the CG, so that the CG
subspaces
efficiently;
(ii) find a
on T1.
to form such a subspace
residual
more efficient,
or an approximate
are not favorable
to solve the subproblem
to
of A on one
an eigenspace
of A on the other
One of the natural choices where
which are not favorable
in the spectrum
In order to make the CG algorithm
T, which contains
solves the subproblems
= 0
are embedded
then they no longer
subspaces.
&k)
&k
3 show that if the eigenvalues
or their approximations subspaces,
other conjugate
ck(
lim E-o
and
generated
T, is to let Z, = [?,,,
by the CG algorithm
. , Pm],
for solving
t E S. There are four advantages to such a natural choice:
ri can be formed throughout
the computations
by their recurrence
relation. (2) The quantities can be computed
ci used to estimate
explicitly
(3) Since T1 = K,_,,
the Krylov subspace
m such that T, contains well-separated
extreme
(4) The matrix performance
an eigenspace
3
of dimension
or its approximation
m, there exists an integer corresponding
to those
eigenvalues.
Zf AZ, is a tridiagonal
is not affected
of A, such as the direct
algorithms
whose
method,
can be
Z: AZ, f(l) = b,.
complement
T, = span Z, = null[( AZ,)‘].
matrix, and many efficient
by the spectrum
adapted to solve the subproblem To form a conjugate
the lower and upper bounds in Theorem
or can be easily estimated.
of subspace
A Householder
T,
with respect
orthogonalization
to A, we may let
procedure
can be ap-
AUBURN 1990 CONFERENCE
ON MATRIX THEORY
743
plied here to form an orthonormal basis which generates the subspace Ts, and the spectrum of A on T, can be estimated by Theorem 3. This decomposition can be carried on again on the subspaces to that a set of conjugate subspaces are formed successively. Observe that if Ta is chosen as an orthogonal complement of Tr, then its basis matrix 2, can be formed as 1 - ZrZi without any extra cost. With this choice of ‘I’,, the solution of(P) is no longer simply the sum of the subsolutions on Tr and Ta, but it can be computed from those subsolutions by using the Sherman-Morrison-Woodbury formula. The complexity and the performance of the algorithm with these choices, and the scheme used to determine the dimension of Tr, are discussed with details in [4].
REFERENCES 1
2
3
4
5
6 7 8 9 10 11 12
0. Axelsson and G. Lindskog, On the Eigenvalue Distribution of a Class of Preconditioning Methods, Group Report 3, Dept. of Computer Sciences, Chalmers Univ. of Technology, Goteborg, Sweden, 1985. 0. Axelsson and G. Lindskog, On the Rate of Convergence of the Preconditioned Conjugate Gradient Methods, Group Report 5, Dept. of Computer Sciences, Chalmers Univ. of Technology, Goteborg, Sweden, 1986. M.-Q. Chen, The Updated Conjugate Subspace Method in Optimization and in Solving Linear Systems of Equations, Ph.D. Thesis, Dept. of Mathematics, Univ. of Illinois at Champaign-Urbana, 1989. M.-Q. Chen and A. Sameh, The Conjugate Subspaces Decomposition and Its Application in Solving Linear Systems of Equations with Many Right-Hand Sides, Tech. Report 1024, Center for Supercomputing Research and Development, Champaign-Urbana, Aug. 1990. S.-P. Han, Optimization by updated conjugate subspaces, in Numerical Analysis (D. F. Griffiths and G. A. Watson, Eds.), Pitman Res. Notes Math. Ser., 1986, pp. 82-97. A. Jennings, Influence of the eigenvahre spectrum on the convergence rate of the conjugate gradient method, J. Inst. Math. AppZ. 20:67-72 (1977). D. P. O’Leary, The block conjugate gradient algorithm and related methods, Linear Algebra Appl. 29:243-322 (1980). C. C. Paige, The Computation of Eigenvalues and Eigenvectors of Very Large Sparse Matrices, Ph.D. Thesis, Univ. of London, 1971. B. N. Parlett, A new look at the Lanczos algorithm for solving symmetric systems of linear equations, Linear Algebra Appl. 29:323-346 (1980). J. &ad, On the Lanczos method for solving symmetric linear systems with several right-hand sides, Math. Cump. 48(178):651-662 (Apr. 1987). A. van der Sluis and H. A. Van der Vorst, The rate of convergence of conjugate gradients, Numer. Math. 48:543-560 (1986). A. van der Sluis and H. A. Van der Vorst, The convergence behavior of Ritz values in the presence of close eigenvahres, Linear Algebra Appl. 88/89:651-694 (1987).
744 INVERSES
FRANK UHLIG, TIN-YAU TAM, AND DAVID CARLSON OF MATRICES
ARISING
FROM
DIFFERENCE
OPERATORS
by SUI SUN CHENG3*
Explicit inverses and inversion algorithms for square matrices have been a major concern since the early days of matrix theory. As the areas of application of matrix theory began to broaden, band and Toeplitz matrices [2] found their way into several methods of numerical analysis and approximation theory. For example, they would arise when using finite difference methods for differential equations, when using polynomial splines, and in studying discrete random processes [3] and statistics [4]. When using finite difference methods, for instance, it is desirable to find accurate error bounds. Explicit inverses or properties such as bounds or asymptotic behavior of the elements of the inverses are needed for this purpose. A common way to obtain this information is to observe that each column of the inverse of, say, a Toeplitz matrix satisfies a linear recurrence equation wth constant coefficients. This recurrence equation can then be solved, at least in theory, and the solution expressed as a linear combination of powers of roots of the characteristic equation associated with the recurrence equation (see e.g. [5, Chapter 41). Thus, the properties of the elements of the inverse can be deduced by this process. As an example, consider the well-known tridiagonal matrix A = (u~~)“~,,, where aii = - 2, aij = 1 if ( i - j 1 = 1, and aij = 0 otherwise. If we denote the jth column vector of the inverse A- ’ by col( x(l), x(2), . . , x(n)), then the components of this vector satisfy
x(k -
1) -
2x(k)
+ x(k +
1) = 6kj,
k=
1,2 ,...)
It,
where we have defined x(O) = 0 and x(n + 1) = 0, and Likj= 1 if i = j and iikj = 0 otherwise. It is convenient to employ the forward difference operator A, defined by Au(k) = u(k + 1) - u(k), to write the above equations as
A%(k
-
1) = i_ikj,
k=1,2
,...,
n,
x(o) = 0 = %(rt + 1). Taking our cue from the theory of Green’s functions in differential would guess that the solution of the above problem is of the form
equations,
we
x(k)=a+bk+(k-j),,
32Department
of Mathematics,
work was funded by the National
Tsing Hua University, Research
Council
Taiwan,
of the Republic
Republic of China.
of China.
This
AUBURN 1990 CONFERENCE
745
ON MATRIX THEORY
where (Y+= Q if cr > 0 and LY+= 0 if (Y< 0. Then it is easily calculated that a = 0 and b = - (n + 1 - j)/(n + 1). By symmetry considerations, we would also guess that
x(k) = c + d(n + 1 - d) + (j - k)+ and deduce that c = 0 and d = - j/(n + 1). By means of the definitions and (j - k) +, we see that x(k) is also given by
of (k - j),
k(n+l-j)
x(k)
=
I
n+l
’
k
j
_j(n+l-k), n+l
This example motivates a generalization for constructing explicit inverses of matrices for which the inner product of its kth row with the jth column of its inverse can be written in the form A”%(k - t) = Skj,
(1)
where m + 1 is the band width and 1 Q t Q m - 1. The details of this generalization will appear elsewhere [l] and will not be repeated here. However, we shall quote the following theorem, which is central to the derivations of explicit inverses. THEOREM. Let 2 4 m < n, 1 4 t 6 m - 1, and 1
k=
. . . . -1,&l
G n. Let P,,,_l be the set of
with degree less than or equal to m - 1. Then x(k), ,... by
+)
p(k)
z
+
defined fw
(k-‘(; t-l;/:“-1), P~P?l-,
or
+)
=
+)
+
(-l)“(j -;m’_ml-,l + #?-“,
satisfies Equation (l), wher-e fw c@ = cY(a - 1) *. 9 (a -p+l) j3 = 0.
any integers if
a>0
9EP,.-,,
(Y and & @) = 0 if (Y < 0 or /3 < 0,
and
/3>0,
and c@)=l
ifa>0
and
We remark that even though explicit inverses can be constructed in principle by the above theorem, we have not derived subsequent properties of the inverses. It will be of interest to find norms or bounds or asymptotic behavior of the elements of the inverses by means of the method mentioned above.
746
FRANK UHLIG, TIN-YAU TAM, AND DAVID CARLSON
REFERENCES 1
S. S. Cheng
and L. Y. Hsieh, Inverses
of matrices
arising from difference
operators,
Utilitas Math., to appear. 2
D. S. Meek, A survey of the results on the inverses of Toeplitz Proceeding of the Conference RIMS,
Implementations, 3
B. N. Mukherjee matrices
W. F. Trench,
5
K. S. Miller,
12:515-522
definite
Toeplitz (1988).
An algorithm for the inversion of finite Toeplitz matrices,
J. SIAM
to the Cakx~us of Finite Differences
An Introduction
and Di&jerential
Dover, New York, 1966.
MINIMAL RANK AND MAXIMAL FOR CERTAIN BAND MATRICES by JEROME
RANK HERMITIAN
COMPLETIONS
DANCIS33
In the last decade Hermitian
of positive
Linear Algebra Appl. 102:21 l-240
applications,
(1964).
Equations,
specified
in
and the
Kyoto Univ., Kyoto, Japan, July, 1982.
and S. S. Maiti, On some properties
and their possible
4
and band matrices,
on Standard Algorithms jw Linear Computation
Hermitian
a popular
matrix either
problem
has been
to a Hermitian
to try to “complete”
matrix with prechosen
a partial
inertia
or to a
matrix with the maximum or the minimum possible rank. The purpose of this
paper is to present band matrices
the standard
together
method
for constructing
Hermitian
completions
of
with some results on minimal and maximal rank Hermitian
completions. The
DEFINITION.
{ rr( H), v(H), 6(H)}
inertia
consisting
ues of H. We let r(H), Our starting
THEOREM 1 [I].
n x n Hermit&
a
Hermitian
matrix
H
of positive,
is
negative,
a
is our generalization
of Poincare’s
triple
In H =
and zero eigenval-
Y(H), and 6(H) denote the three coordinates
point
interlacing theorem
of
of the numbers
of In H.
inequalities
and Cauchy’s
(notation: 0’ = 0 E Cr):
Let R, be the leading principal
matrix
H.
(n -
r) x (n - r) submatrix
of an
We set A = Dim Ker 23, - Dim[(Ker R,) @ Or] fl Ker H.
Then (a) 7r( H) 2 T( R,) + A and (b) r(H) Clearly,
> r(R,)
+ A.
the inertia
of every
Hermitian
completion
of a (band) matrix
must be
consistent with this theorem.
33Department of Mathematics, University of Maryland, College Park, MD 20742.
AUBURN 1990 CONFERENCE
ON MATRIX THEORY
747
BORDERED-MATRIXHYPOTHESES. Let H, be an (r - 2) x (r - 2) Hermitian matrix. Let o and w be vectors in C’-‘, and let a and b be real numbers. Let H(z) be the bordered matrix
H(Z)
=
a v
::
.z* w ,
Iz
w*
b 1
and let
and
H, =
H(z) is called a one-step compkiun of H(0). The one-step-completion problem for these bordered matrices is usually the inductive step in the proofs of completion theorems. We used Theorem 1 as we classified all the possible kernels of bordered matrices in Lemmas 3.3-3.9 of [2]. A sample of the “common threads” of these lemmas is Theorem 2. THEOREM 2 (Theorem 1.2 of [2]). Given the bordered-matrix hypotheses. Suppose that the nullities are non-decreasing, namely,
6(H2)
and h(h) 6
B(h)
(that is, the s&matrix does not have a larger nullity). Then there is a number z. such that there is a preservation of positivity and negativity, namely,
“(H(Q)) =M~{+++G)j
and +(G))
= M~{~(H+OS))
and again the nullities are non-decreasing, namely,
6(HI) g 6(H(G)) (that is, the new s&matrices
and 6( Hs) 6
h(H( ~0))
do not have larger nullities).
DEFINITION. A matrix with all zeros off the main diagonal and the first m pairs of superdiagonals is called an m-band matrix, that is, R = (rjk) and rjk= 0 for all (k - j ( > m. An n X n Hermitian matrix F = (f$) is a compktion of an m-band matrix fl if fjk = rjk for all 1k - j ) < m. The maximal Hermitian (m + 1) x (m + 1) submatrices R,, R,, . . . , R,_, within an m-band n x n matrix R are Ri = ( rjk Ij,k = i, i + 1, . . . , m + i). The almost maximal Hermitian m x m submatrices of R are RT = (rjk (j,k = i, i + 1, ...,m + i - 1). RT, R;, . . . , &,,+I
748
FRANK UHLIG, TIN-YAU TAM, AND DAVID CARLSON A simple diagonal completion R’ of an m-band Hermitian
DEFINITION. an m + l-band
Hermitian
of an m-band
completion
(N - 1)st (successive)
Hermitian
as the s&matrices
hypotheses.
Also,
completion
matrix
An
N th (successioe)
R is a simple
H,,
H,,
and
H,,
completion
Ri and Ri+l
of an
and their “com-
of the bordered-matrix
respectively,
the ith maximal
the maximal
almost maximal s&matrices
diagonal
R is
diagonal
RT of a band matrix R always overlap in the same
H( .zo) becomes
R’. In addition,
matrix
simple
of R.
Each pair of maximal submatrices
almost maximal submatrix
manner
of R.
simple diagonal completion
OBSERVATION 3. mon”
completion
of the simple
submatrix
submatrices
diagonal
of a band matrix R become
the
of its simple diagonal completion R’ (with the same ordering
from the upper left corner). STANDARD METHOD (For matrices). R’, then
Complete complete
diagonal completion completion
R’ to R” =
completions will construct Theorem
a Hermitian 4 below
of our
and negativity
completion
3, and the Standard
Method
rank Hermitian submatrix
completion
and 6(Ri+1)
property.
will establish completion
of a Hermitian
= Max{*(Ri),i
method”
As an example, the next theorem.
result.
It says that
band matrix
R is not
then there is a positivity
Given a Hermitian
RI, R,,
m-band matrix R with
. . , R,_, and R:, RX,. . , R*,-,+l.
namely, foreach
i=1,2
,...,
n-m-l.
F of R such that there is preservation
= 1,2 ,...,
= Max(v( A,), i = 1,2,. (We announced
of
completions
the “standard
namely, z(F)
kernels
on one-step
of R.
2 6(Ry)
completion
possible
By using these one-step
of its pair of almost maximal submatrices,
Then there is a Hermitian
simple diagonal
theorems
completions,
with the desirable
Suppose that the nullities are nondecreasing,
and negativity,
of the
properties.
THEOREM 4 (Minimal rank completions).
> 6( Ry)
classification
simple diagonal
is a minimal
preserving
A to
completions.
desirable
maximal and almost maximal submatrices
6(R,)
3, each successive
one-step
of [Z]), one may obtain
the nullity of each maximal
less than the nullities
n x n
complete
F is an (n - m + 1)st simple
3.3-3.9
of successive
2, Observation
Theorem whenever
threads”
(Lemmas
which propagate
in the construction
of m-band
namely,
is the method used in [3-71.
“common
matrices
completions
(R’)‘. . . to F, where
by a set of independent
This standard method bordered
Hermitian
simple diagonal completions;
of R. After noting Observation
is achieved
By finding
constructing
successive
n - m}
and
V(F)
. . , n - m).
this result at the ILAS meeting
in Provo, Utah, 1989.)
of positivity
AUBURN 1990 CONFERENCE
749
ON MATRIX THEORY
Of course, the inertia of a Hermitian completion F of a partial Hermitian n X n matrix R must be consistent with Poincare’s inequalities and Cauchy’s interlacing theorem, that is, for each maxim+ specified principal submatrix R, of R, r(F) ) *( RJ and V(F) ) v( Ri). Therefore Theorem 4 provides the minimal possible rank among all possible Hermitian completions. Dym and Gohberg presented the standard method in [5] as they showed that, when all the maximal submatrices of a Hermitian band matrix R are positive definite, then R has a Hermitian completion F which is also positive definite, such that F-’ is also a band matrix. They report that this result has connections with signal processing and system theory. Johnson and Rodman have shown (in [7]) that when all the maximal submatrices of a Hermitian band matrix R (or even more generally a matrix with a “chordal” graph) are invertible, then R has an invertible Hermitian completion. Ellis, Gohberg, and Lay have shown (in [S]) that when all the maximal submatrices and all the almost maximal Hermitian submatrices of R are invertible, then R has an invertible Hermitian completion F for which F-’ is also an m-band matrix. THEOREM 5 [3]. < n - m, suppose
LetRbeanm-bandn~nmutrix. that R,
Furanintegerr,m+5<2r
or R’f is invertible.
Then R has an invertible
Hermitian
completion. COUNTEREXAMPLE 6. An example of a l-band 3 x 3 matrix, with an invertible maximal Hermitian submatrix, which does not have an invertible completion is
1
010 1 01
0 0.
I
Theorem 7 is a generalization of Theorem 5, but its Hermitian or may not be the one with the maximal possible rank.
completion
F may
THEOREM 7 [3]. Let R be a Hermitian m-band n X n matrix. Suppose that m 6( R,) + 5 Q 2 r < n - m for some integer r. Then R has a Hermitian completion F with S(F) Q 6(R,).
DEFINITION. Given an m-band n x n matrix R = (rij), its maximal full-column is the unique specijied n x (2m + 2 - n) submattix
submatrix
M=
(rij(i=
1,2,...,
r
and
j=n-m,n-m+l,...,
m+l).
REMARK. Since the columns of M will also be columns of any completion Ker F > On-m-1
e Ker M e O”-m-1
and
RankF,<2(n-m-l)+RankM
F of R,
750
FRANK UHLIG,
for any completion matrix,
TIN-YAU TAM, AND DAVID CARLSON
F of R. In the trivial case, when 2m + 2 < n and M is an empty
we set Ker M = 0 and Rank M = 0.
provided by Theorem
Therefore
the
Hermitian
completion
8 has the maximal possible rank among all (including non-Hermi-
tian) completions. 8 (A maximal rank, minimal kernel completion
THEOREM
n x n Hermitian matrix R with maximal s&matrices IRank Ri - Rank Ri+l
1< 1
forall
R,, R,,
[4]).
1,2 ,...,
i=
Given an m-band
, R,_,.
.
Suppose that
n-m-
1.
Then for almost all Hermitian completions F of R, Ker F = O”-m-1
@ Ker M @ On-m-1
Rank F = 2( n - m - 1) + Rank M,
and
where M is the full-column maximal submatrix
of R. if R is also a real symmetric matrix, then the conclu.sions are valid for all but possibly a finite number of the real symmetric completions. Furthermore,
Curiously, Theorem
REMARK. demonstrated
8 is not valid for non-Hermitian
band matrices,
as is
by this l-band 3 x 3 matrix:
F( n, y) = 1 Here Rank F( x, y) = 2 # 3 = 2(3 -
1 -
0 0
1 0
x 0
Y
1
0
. i
1) + Rank M for all values of x and y, even
though Rank R, = Rank R, for the only maximal specified submatrices.
REFERENCES Jerome
Dancis,
operators,
On the inertias
of symmetric
Linear Algebra Appl. 105:67-75
Jerome
Dancis, Bordered
Jerome
Dancis,
matrices,
Invertible
matrices
and bounded
Linear Algebra Appl. 128:117-132
completions
self-adjoint
(1988).
of band matrices,
(1990).
Linear Algebra Appl., to
appear. Jerome Dancis, Minimal Rank and Maximal Rank Hermitian Completions Certain Band Matrices, Technical Report Tr 89-58, Univ. of Maryland, 1989. Harry Dym and Israel Gohberg,
Extensions
of band matrices
for
with band inverses,
(1981).
Linear Algebra Appl. 36:1-24 Robert L. Ellis, Israel Gohberg,
and David C. Lay, Invertible self-adjoint extensions
of band matrices and their entropy,
SIAM J. Algebraic
Discrete Methods 8:483-500
(1987). Charles
R. Johnson
partial Hermitean
and Leiba
matrices,
Rodman,
Inertial
Linear and Mu&linear
possibilities
for completions
Algebra 16:179-195
(1984).
of
AUBURN
ON CHARACTERIZATIONS OPERATORS by
THEORY
~~~~C~NFERENCEONMATRIX
OF THE SPECTRAL RADIUS OF POSITIVE
SHMUEL FRIEDLAND Let B be a Banach space over the real numbers
the Banach space of all real-valued pointed
(K fl - K = (0))
function&
with
cone.
respect
to
K.
bounded Denote
then let
x ( y iff
B = K - K and bounded
K* f (0).
in norm, then
linear operator.
Denote
Note
let
y 2 x and
Furthermore,
interior,
called
positive
if
Collatz-Wielandt
AK C K. Assume
v(A)
sets associated
-e
: B + B be a bounded
the local spectral
radius of A at x.
linear operator
A is a positive
operator,
A : B + B is
and define
= (o:3r>O,
Ax
X1(A)
= {a:3x%+O,
Ar
“(A)
= {w:3x>O,
Ax>wr},
$(A)
= (w:3r+O,
Ax>wx}
the upper and lower Collatz-Wielandt =inf{a)O:Ar
R(A,x)
the
with A [2]:
“(A)
For xcK
that
Q x < e is
= mimi)lhl.
p( A, r). A bounded that
A
by K,.
implies
of A. Set
For x E B let p( A, x) = lim sup I( A’% ((‘lrn denote ,tal
K, # 0
the segment
B* = K* - K*. See [7j or [S]. Let
linear
y - r E K and
and denote its interior
if for eEK,
by a( A) the spectrum
T c B set V( A, T) = inf,..r
by B*
K C B be a closed
y > x iff
that the assumption
P(A) = pan, I XI,
For
]] I]. Denote
Let
by K* C B* the cone of nonnegative
As usual,
y - XEK,.
with the norm
linear function&.
y - r E K \ (0). Assume that K has a nonempty We
751
numbers
are defined as follows:
r( A, x) = sup{w
> 0: Ax > UK}.
(1)
Note that R( A, r) = 00 if no u exists such that An < UT. Clearly,
sup Qr( A) = :“,“or( A, x) G sup Q( A) = supr( X>O infX(A)
In the case of B = numbers
= mf,R(A,x)
R”, K = R;, and A an irreducible
A, x),
\
= I=f,R(A,x).
nonnegative
matrix, all the above
are equal to p(A). This is the classical result due to Wielandt.
any finite-dimensional
34Department
B with a closed
of Mathematics,
pointed
spanning
cone
I
It also holds for
K and a positive
University ofIllinois at Chicago, Chicago, Illinois 60680.
752
FRANK UHLIG,
irreducible
operator
A. However,
TIN-YAU TAM, AND DAVID
if A is not irreducible
have to be equal even in the finite-dimensional following theorem
characterizes
local spectral
do not
[6] and [9]. The
in terms of the spectral
radius
radii of A and A* [5]:
Let B be a real Banach space with a closed pointed cone K. Assume interior and B* = K * - K*. Let A : B + B be a
THEOREM 1. bounded
the above four numbers See for example
the above four numbers
and the minimal distinguished
furthermore
case.
CARLSON
that K has a nonempty
linear operator which leaves K inuariunt.
Then
(3) ji$R(
A, x) = v( A, K),
sup r( A, x) = r~( A*, K*),
(5)
x*0
supr(A,x)
x>o Zj p(A) i.s in Recall pointed;
of A, then
the point spectrum
that
a finite-dimensional
Let C be a C*-algebra. of the form aa*. Let
Denote
K
is generating
B Theorem
(B = K - K)
of K, i.e.,
iff
is
K*
1 is due to [9].
by K the cone of all self-adjoint
be the interior
K,
equality holds in (6).
cone
e.g. [l]. For a finite-dimensional
(6)
(positive)
elements
K, is the set of invertible
elements
in K. THEOREM 2 [5].
Let C be a C*-algebra.
positioe linear operator p(A)
Assume
that A : C -+ C is a bounded
(AK C K). Then
= ,i{
u( A*, K*)
@‘Ax),
= sup v( &4x).
0 The above theorem tion.
Indeed,
product
on
self-adjoint
can be considered
C: (a, b) = +(ab*),
QC Endz(C).
as an extension
if C is a finite-dimensional
positive
elements
where
of C, e.g.
Note that the subalgebra
$? is a semisimple
algebra.
(7)
EK”
C* algebra, 4
is a positive
[8]. Then
functional
characteriza-
on the
C can be viewed
@? is invariant
More precisely,
of Wielandt’s
then one can define
vector
cone
of
as a subalgebra
under the involution
the underlying
an inner
space
*. Hence,
C splits to a
direct sum VI + * * * + Uk where each Vi is an irreducible invariant subspace under the action of ‘iR. It then follows that the restriction of V to uj is isomorphic to M,,(C), where mi = dim Vi. Thus, any finite-dimensional
C*-algebra
c = Mm,(C) + *. * +%f,,(c) .
is isomorphic
to
AUBURN
1990 CONFERENCE
ON MATRIX
753
THEORY
Here,
a=
( a,,...,
Q),
aiEMmi(C),
@I,..
i = l,...,
.>a#$.
.>bk) = (a$,,..
Thus, if A : C + C is a positive operator elements,
ly,ykP(x;‘( Ax)i),
a;)>
.dQbk)>
to the cone of self-adjoint
v( x-'Ax)
. .
(xl,. . ., xk),
Ax = ((Ax)~,..
Note that C is isomorphic C is a commutative i = 1,.
with respect
a* ( r,...,
That
under the pointwise
is, in (8) we have
characterization
C = M,(C)
for a nonnegative
and a positive
Consult
irreducible
operator
[4] for other characterizations
with respect
to general
irreducible
Ax)i),
i = 1,. . .,k.
xi = aia’,
.,(Ax)~),
to Ck as a C*-algebra
C*-algebra.
mh,v( x;l(
=
the
matrix
(9)
multiplication
equalities
. . , k. As p(a) = Y(U) = a for 0 < a E M,(C) = C, we deduce
Wiehmdt
positive
we deduce
P( x-‘Ax) = LX=
a* =
k,
iff
ml = 1,
that (7) is the
A. Theorem
2 for
A was proven already in [4].
of the spectral
radius of positive
operators
cones, and [3] for related results.
REFERENCES A. Berman
and R. J. Plemmons,
Academic,
New York,.1979.
G. P. Barker and H. Schneider,
Appl. 11:219-233 K. H. Fijrster
S.
matrix,
Characterizations
Friedland,
theory,
numbers
Linear Algebra
and the local spectral
Linear Algebra Appl. 120:193-205 of the spectral
radius of positive
(1989). operators,
Linear
operators
on
(1990).
Characterizations
of spectral
radius
of positive
C*
/. Funct. Anal., to appear.
S. Karlin, Positive
operators,
J.
M. G. Krein and M. A. Rutman,
(1959).
Math. Mech. 8:907-937 Linear operators
space, Uspekhi Mat. Nauk 3:95 (1948); G. K. Pedersen,
leaving invariant cone in a Banach
Amer. Math. Sot. Transl.
No. 26.
C*-Algebras and Their Automorphism Groups, Academic,
B. S. Tam and S. F. Wu, preserving
Perron-Frobenius
and B. Nagy, On the Collatz-Wielandt
Algebra Appl. 134:93-105 algebras,
Algebraic
(1975).
radius of a nonnegative S. Friedland,
Nonnegative Matrices in the Mathemutical Sciences,
On the Collatz-Wielandt
map, Linear Algebra Appl. 125:77-95
sets associated
(1989).
1979.
with a cone
754
FRANK UHLIG,
TIN-YAU TAM, AND DAVID
EXTREME DOUBLY NONNEGATIVE
CARLSON
MATRICES WITH PRESCRIBED
ROW SUMS by ROBERT GRONE35
Let either
Zf, denote
the convex
the real or the complex
extreme
cone
of all n-by-n
positive
semidefinite
case, it is obvious that a matrix
ray in H, if and only if the rank of A is 1. Several
structure
of the set of extreme
example,
the
diagonal. points,
correlation
The extreme have been
set the notation,
for example,
rays of subcones that
familiar convex
convex
(undirected,
interest
set of matrices
set in
obtained
by establishing
is of some
H,
matrices
matrices,
the ranks of between
the
this result being
and methods
interest.
if p = (p,,
employed K,
role.
previously
One useful
Our
in describing for simplicity,
mentioned
concept
a lot of
that it held for A in K,, techniques
extreme
be the set
. , p,. Although points of K,
and because
of extreme graph
are
some of the
points will be similar
in that rank and iero patterns
will be the
so
are valid in
> 0, let K(p)
. . . , pJT
papers deal only with K,. The investigation
the extreme
stimulated
in H, which have row sums pr,
we will use the notation
to that of the other problems an important
To
is a,,, the set of n-by-n doubly stochastic
conjecture
In particular,
nonnegative
valid for K(p),
authors
with no loops or
and there is a distinction
points of 62, are just the permutation
more generality.
of all entrywise
referenced
various
a given sparsity pattern.
some 40-odd years ago. In this note we wish to consider
this particular
For
on the
. . , n}. and let M(G) consist of all A in H, such that
activity on the van der Waerden
the theorems
l’s
ranks of extreme
[7], and [lo]
points of the set K, = a2, fl H,. In 1962, Marcus and Newman [ll]
somewhat
of H,.
have
cases.
The extreme
due to Birkhoff
subsets
which
i # j and (i, j) is not an edge in E. In these investigations
real and complex Another
authors have studied the
the possible
in [9]. In [l],
of H, which respect
points or rays are of particular
matrices.
In an
convex
in H,
C = (V, E) is a simple
multiple edges) graph on V = (1, extreme
matrices
points of this set, and in particular
suppose
aij = 0 whenever
or rays of certain
are those
matrices
studied,
studied the extreme
points
matrices.
A in H, generates
of a symmetric
play
matrix.
If
A = AT is n-by-n, we let G(A) be the graph on V = { 1,. . , n} with edges (i, j) for all i, j for which V = (1,.
i # j and aij # 0. If G, = (V, E,)
. , n}, we say that G, is a s&graph
we use the notation
and Fisher
in K,.
They established
This work
DMS-9007048.
matrix in K,
in H,. Furthermore,
35Department 92182.
noted that I, is always an extreme
in Q,, and that I,, is the only extreme
noted that the unique rank-l it is extreme
In this case
G, C G,.
In [5], Christensen it is extreme
and G = (V, Ea) are two graphs on
of G, if and only if El C E,.
(l/n)J,
has been
Sciences,
supported
since
They also
is (l/n)J,, and that it is extreme in K, since is the only entrywise positive extreme point
also that rank-2 matrices
of Mathematical
point in K,
point of rank n in K,.
are extreme
San Diego
by the
National
State
in K,
if and only if they
University,
Science
San Diego,
Foundation
under
CA grant
AUBURN
1990 CONFERENCE
have zero entries.
These
the authors
obtained
established
an inequality
ON MATRIX
observations
tell the complete
some tridiagonal
extreme
which relates
755
THEORY story for n = 2,3.
For n = 4,
points of rank 3. For general
the number
of nonzero
entries
n, they
of an extreme
point with its rank. These
results
determine
suggest
extremality
the possibility
for A in K,.
the rank of G be the minimum
that rank A and G(A)
If G is a connected
rank of A in K,
might suggest that for a given G, either no matrices or else the extreme
matrices
in K,
with G(A)
might be sufficient
graph on V = (1,.
with G(A) in K,
= G. The results
with G(A)
= G are exactly
to
. . , n}, let so far
= G are extreme,
those with rank A =
rank G. We shall see later that this is not the case. In testing any A E K, matrix C a perturbation
for extremahty,
the following notion from [8] is useful.
Call a
of A if and only if
(i) C = CT, (ii) Ce = 0,
A C nullspace C, and
(iii) nullspace
(iv) G(C) C G(A). These four constraints perturbation K,
space is trivial.
is replaced The
are linear and homogeneous,
of the n-by-n real matrices.
a subspace
A matrix
results
rank A = n criteria
of A form
The reader should note that this definition
is unchanged
in [5] give a characterization 1. We assume without
that
C(A)
of the extreme the structure
loss of much generality
is connected.
By counting
points
of extreme that
equations
A in K, A in K,
rank A = n -
with
and unknowns
or
in the
to exist, we saw that if rank A = n - 1 and A is extreme,
for a perturbation We established
with
A is irreducible,
then G(A) has less than n edges. This forces G(A) to be a tree, or equivalently, acyclic.
if
by K(p).
rank A = 1, 2, or n. In [8], we considered equivalently,
and so the perturbations
A in K, is extreme if and only if this
that when
G(A)
is a tree,
then
A is extreme
1. Also, if rank A = n - 1 and A is irreducible,
A to be
if and only if in K,
then A is extreme
if and only if G(A) is a tree. In [2] a lemma
was obtained
which
was also useful in [7j and [8]. The matrices
studied in [2] were doubly nonnegatiue, that is, positive semidefinite nonnegative. irreducible enabled
Clearly,
in K,
the matrices
and doubly nonnegative,
and G(A)
us in [8] to give a complete
bipartite.
and
K(p)
is bipartite,
answer to when
if A in K, has G(A) bipartite
Specifically,
as well as entrywise
fall into this category.
If
A is
then rank A 2 n - 1. This
A in K,
is extreme
and connected,
if G(A)
is
then A is extreme
ifandonlyinG(A)isatreeandrankA=n-1. In [8] we also considered exactly
the case when G is unicyclic
n edges and one cycle).
bipartite
and there
is no extreme
If the unique
(that is, G is connected
cycle in G is of even length,
with
then G is
A in K, with G(A) = G. If G has a cycle of odd
length, then any A in K, with G(A) = G has rank at least n - 2. In this case, then A in K, with G(A) = G is extreme if and only if rank A = rank G = n - 2. We also established
the existence
rank, n - 2. Showing result
for K(p)
of A in K,
which has this graph and also has the minimal
that this minimum
by invoking
rank is obtained
the DAD theorem
[4]. Lastly,
in K,
also establishes
the
we noted that if G is any
756
FRANK UHLIG,
graph on V = (1,.
TIN-YAU TAM, AND DAVID
. . , n} for which there is a matrix
A in K,
rank A = k, then for every m, k < m 4 n there exists a matrix andrankB=m. In [6], we considered Say that agraph common
edge.
a class of graphs which generalizes
G on V= Suppose
{l,...,
n} is nonchordal
G is a connected
cycles
and
nonchordal
has exactly
r even
s odd cycles.
corresponds
to a tree, and that r + s = 1 corresponds
results in [8] can be used to establish
an induction
that rank G = n - s - 1, and we constructed minimal
rank. Again, the DAD theorem
has G(A) nonchordal
connected
such a matrix is extreme
Berman
graph on V = (1,.
. , n} which
that the case
in K,
example
points
points in K, lays to rest
qualitatively Lastly,
when
r = s = 0 Hence
with specified
this to K(p).
Suppose
and s odd cycles.
that rank A and G(A)
for n = 4. For completely
n = 5 this pattern
determine
different
possibility
of a nice,
points
in K,.
for certain
p.
the presentation
in [l], [7j, and [lo]. of these
exactly the same if K,
results
characteriIn [3],
of two matrices
simple
answer
investigations
to the K,
problem
and
are closely
the extremality
in K,
and the other not. This
As R. Brualdi quite correctly
in Auburn,
is replaced
1.
to determine
fails to hold.
It may even turn out that
we would like to note that these
sparsity questions
A in K,
We found that
the rank and sparsity possibilities
for n = 5. They also give an example
the
extreme
the
graph and
that
are sufficient
with equal ranks and the same graph, but where one is extreme characterizing
graphs.
of G share a
on r + s. For such a graph we found
with r even cycles
suggests
and Shaked-Monderer
of extreme
trees and unicyclic
= G
if no two cycles
This is true for n < 4, and our results in [6] provide a complete
zation of the extreme
= G and
to G being unicyclic.
matrices
extends
with G(A)
B in K, with G(B)
if and only if r = 0 and rank A = rank G = n - s -
So far the evidence extremahty.
Note
CARLSON
related
pointed
question
K(p)
of are
to the
out during
for A in K,
is
with K, Il M(G( A)).
REFERENCES J. Agler, J. W. Helton,
S. McCullough,
and L. Rodman,
Positive
definite
with a given sparsity pattern, Linear Algebra Appl. 107:101-149 A. Berman and R. Crone, Bipartite completely positive matrices, Philos. Sot. 103:269-276 A. Berman
(1988).
and N. Shaked-Monderer,
More
doubly stochastic matrices, preprint. R. Bruahh, The DAD theorem for arbitrary 45:189-194 J.
points,
R. Grone, matrices, R.
Grone
on extremal row sums,
positive
Proc.
semidefinite
Amer.
Math. Sot.
(1974).
Christensen
extreme
matrices
(1988). Proc. Cambridge
and P. Fisher,
R. Loewy,
and S. Pierce,
Linear and Multilinear and
Positive
Linear Algebra A&.
S.
Pierce,
131:39-50 (1990). R. Grone and S. Pierce,
Extremal
doubly (1986).
Nonchordal
positive
stochastic
matrices
semidefinite
and
stochastic
Algebra, to appear.
Extremal
Linear Algebra Appl. 150:107-117
definite
82:123-132
bipartite
positive (1991).
matrices,
semidefinite
Linear
Algebra
doubly stochastic
Appl.
matrices,
AUBURN 9
1990 CONFERENCE
R. Crone,
ON MATRIX
757
THEORY
S. Pierce,
and W. Watkins, Extremal correlation matrices, Linear (1990). J. W. Helton, S. Pierce, and L. Rodman, The ranks of extremal positive semidefinite matrices with given sparsity pattern, SZAMJ. MatrZx Anal. 10:407-423 (1989). M. Marcus and M. Newman, Inequalities for the permanent function, Ann. of Math. 75:47-62 (1962). B. Ycart, Extremales du cone des matrices de type non negatif a coefficients Algebra Appl. 134:63-70
10 11 12
positifs ou nuls, Linear Algebra AppZ. 48:317-330
RECENT
RESULTS
by SHU-AN
ON THE
(1982).
PERMANENTAL
HU36 and TIN-YAU
NUMERICAL
RANGE
TAM37
The purpose of this synopsis is to give an account of recent advances on the permanental numerical range. The kth permanental numerical range of A E enxn (the set of n x n complex matrices),
where 1 < k < n, is defined as
Pk( A) = {per(U*AU)IUe$,,k,
U*U = Zk},
and per B = CoeS, Htr bj,(i) is the permanent function for BE @kxk. This definition is motivated by the classical numerical range of A E B,xn:
W(A)
If k = 1, then Pr( A) = W(A). upon five properties
= {x*Axlz~V,
IIxII = 1).
There are many nice results for W(A).
here and investigate generalizations
We shall touch
to the permanental
numerical
range. Maybe the most interesting W(A)
one is the. celebrated
Toeplitz-Hausdortf
theorem:
i.sconoexfwanyA~@&,.“.
In particular,
there is a complete
Let A E @2zx2 have eigenvahm
description
Xl and &.
for the shape of W(A)
Then W(A)
when n = 2:
is an elliptical disk with foci
at A, and &,, minor axis of length
dtr(A*A) -
l&l’-
l&l’,
3sDepartment of Mathematics, University of Connecticut, Storrs, CT 06269. 37Department of Algebra, Combinatorics 5307.
and Analysis, Auburn University, AL 36849-
758
FRANK
UHLIG,
TIN-YAU TAM, AND DAVID
CARLSON
and major axis of length
Vtr( In particular,
A*A)
- 2 Re( A,&)
if
c
Al o
A=
hz>
[
1
A, and &,
Recently,
the authors
[5] obtained
Let ~~~~~~
THEOREM 1.
the following
have eigenvalues
minor axis of length
1c 1, and
analogy for Ps( A):
A, and AZ. Then Pz( A) is an elliptical
disk with foci at X,X, and i( XT + hi), minor axis of length
d[tr(
A*A)
-
] X, ] ’ -
] X, ] “] [tr( A*A)
- 2 Re( A1x2)]
,
and major axis of length
tr( A*A)
In particular,
-
IhI2
2
-
-
l&l2
if
Al
A=
0
c
then P2( A) is an elliptical disk with foci at AlA2 IA,-X,12+
= (0)
In [8] Marcus
ifand
W(A)
Recently
Jc12+
)A,-A212/2.
enjoys is:
only ifA
= 0.
and Wang asked whether
they posted it as a conjecture.
THEOREM 2.
and +(A; + A$, minor axis of length
Ic(2,andmajoralrisofZength
Another property W(A)
1
%>
[
ICI
- 2Re(Xr&).
2
Let A E G,,,,
Chan [l] extended
Then
the above property
is true for Pk( A), and
Hu [4] proved the conjecture
1 < k Q n. Then Pk( A) = (0)
the result to arbitrary
ter. Let 1 ,< k < n, and let G < Sk be a subgroup
subgroups
ifand
in the affirmative: only ifA
with principal
of the full symmetric
= 0. charac-
group Sk of
AUBURN degree
1990 CONFERENCE
k, and x
759
ON MATRIX THEORY
: G -+ G an irreducible
character
of C. Then define the following
range:
P,“(A)
= {~;(U*AU)IUEC”,~,
where d:(B) = C oeC x(u)H!=, bio(i), associated with x. Chan [l] obtained:
for
BE
U*U = Zk},
ejkxk~
is the generalized
matrix function
THEOREMS (a) Zfx 3 1, then P:(A)
= {0}
ifand
only ifA = 0.
(b) Zf x f 1, then rank A < 1 implies P,“( A) = (0). The third property
we want to discuss is:
W( A) is a line segment if and only if 5 A is Hermitian for some .$ E G with
1[ 1 = 1.
In [5] the authors obtained a similar result for Pk( A):
LetAEGnx,,.
THEOREMS.
(a) Zf n = k = 2, then Pz( A) is a line segment if and only if A is normal. (b) If 1 < k < n, excepting n = k = 2, then the following statements are equivalent: (i) Pk( A) is a line segment. (ii) Pk( A) is a line segment on a ray containing (iii)t A is Hermitian for some nonzero
COROLLARY
the origin.
t E G with
1.$1 = 1.
&AE@$,,,.
1.
(a) Zf n = k = 2, then P2( A) lies on the real axis (nonnegative real axis; positive real axis, respectively)
if and only if
(i) A or iA is Hermitian ;)
(A or -A
is positive semidefinite;
positive definite, respectively),
A is uniturily similar to diag(a + bi, a - bi) or diag(a + bi, - a + bi), where a and
b are
nonzero
real
numbers
(A is unitarily
similar
to diag(a + bi, a - bi),
where
a2 - b2 > 0; a2 - b2 > 0, respectively). (b) Zf 1 < k Q n, excepting n = k = 2, then Pk( A) lies on the real axis (nonnegative real axis; positive real axis) if and only if 5 A is Hermitian for sollze 5 E G such that tk = 1 or - 1 (.$ A is positioe semidefinite;
positive definite, fm some k th root of unity
.$, respectively). The fourth property
is:
Zf A is positive semidefinite, largest eigenvalues
as endpoints.
then W(A)
is a line segment with the smallest and the
FRANK UHLIG,
760 It is a long-standing semidefinite,
of U*AU are equal.
[lo]
to find max{
min{ 1z 1 I .z EP,( A)}
whereas
Mehta [9] conjectured conjecture,
problem
TIN-YAU TAM, AND DAVID
that the maximum Recently
CARLSON
1z 1 ) z E P,( A)} if A is positive
= det A is well
known
(see
e.g.
is attained when all the main-diagonal
Drew and Johnson
[3] gave a counterexample
[lo]). entries
to Mehta’s
and in [2] they proved that:
THEOREM 5. If A is a 3 x 3 real positioe semideftnite matrix, then max{ I .zI I z E P3( A)} is always of the form $A,( A! + A:) or $[3Ai(h, + A,) + A\ + A” + (% + A\ Ai)3/2] for sine ordering of the eigenvalues.
It turns out that there exists a persymmetric the maximum persists
(i.e.,
symmetric
with respect
to the
diagonal as well as the main diagonal) matrix U*AU that yields
upper-right-to-lower-left
for n = 2,3.
It was then asked in [3] whether
the persymmetry
criterion
for n 2 4.
Lastly, in [ll]
Pellegrini proved that:
A linear operator T: G,x, -+ Ctnxn satisfws W(T( A)) = W(A) and only if there exists U E U,(G) such that
for all A E enxn
if
(i) T(A) = U*AU for all A E Gnxn or (ii) T(A) = U*ATU for all A E GnX,. In [5, 121 the authors extended
the result.
Let T: Gnxn + Gnxn be an operator. THEOREM 6. Pk( A) for all A E C?,x, if and only if: Case I.
Then T satisfis
If 1 6 k < n, excepting k = n = 2, there exist U E U,(c)
Pk(T( A)) =
and a k th root of
unity 5 such that either (i) T(A) = f;U*AU for all A E Gnxn, or (ii) T(A)
= tU*ATUfw
If k = n = 2, there exists UE U,(G) such that either
Case 2. (i) T(A)
all AE@,~,,.
= ~U*AU~~~U~EAE@&~,,
(ii) T(A) = +U*ATUfor (iii) T(A)=
+-I ‘iA
(iv) T(A)=
+ y’l+““*(
REMARK.
2 +iU*
Recently,
or
all AE@&~~, ( AATLei
or
CA 2
z)
yIS)
[7] obtained
UforallAEGZxZ,
or
UforaZlAEG2x2. part
of Corollary
1 independently.
recently the authors [6] extended Theorem 4 and Corollary 1 for arbitrary Sk with principal
More
subgroups
of
character.
REFERENCES 1
C.
T.
Chan,
M&linear
Some
more
on a conjecture
Algebra 25:101-105
(1989).
of Marcus
and Wang,
Linear
and
AUBURN 2
1990 CONFERENCE
J. H. Drew and C. R. Johnson,
The maximum
semi-definite matrix, given
eigenvalues,
25:243-251 3
J.
H.
4
the
and C.
R. Johnson,
permanental
(1989). S. A. Hu,
On
20:191-196
(1987).
the
Counterexample
and
of a 3-by-3
positive
M&&near
Algebra
Marcus-Wang
conjecture,
S. A. Hu and T. Y. Tam, Operators
6
line, to appear. S. A. Hu and T. Y. Tam,
to a conjecture
Linear and Mu&linear
maximization,
5
character,
permanent Linear
(1989).
Drew
regarding
761
ON MATRIX THEORY
Linear
with permanent
On the generalized
of Mehta
Algebra 25:253-254
and M&linear
numerical
numerical
Algebra
ranges on straight
range with principal
preprint.
Tian-Gang Lei, On the numerical M. Marcus and B. Y. Wang, Mu&linear Algebra 9:111M. L. Mehta,
range of an induced power, preprint.
Some variations
on the numerical
range,
Liner and
120 (1980). Hindustan
Elements of Matrix Theuy,
Publishing,
Delhi, 1977,
p.
156. 10
H.
Mint,
21:109-148 11
Theory
of permanents,
V. J. Pellegrini, Numerical T. Y. Tam,
COMPLETELY
KOGAN3’
applications
range of induced
MATRICES
Algebra
n x m matrix.
positive matrices
include
on a Banach algebra, St&a
power,
Linear and Multilinear
AND GRAPHS
and ABRAHAM
is called the factorization
Completely
operators
BERMAN3”
3g
A is completely positive if it can be decomposed
B is a nonnegative
factorization
and M&linear
(1988).
POSITIVE
An n x n matrix where
range preserving
On the numerical
Algebra 23:207-211
by NATALIA
Linear
(1975).
Math. 54:143-147 12
1982-1985,
(1987).
“a proposed
sectors of the U.S. economy”
The minimal number
as A = BB’,
m that admits such a
index of A and is denoted by q(A).
are important in the study of block designs [q. Other mathematical
model of energy demand
for certain
and statistics [5].
A matrix which is both elementwise
nonnegative
and positive semidefinite is called
doubly nonnegative. It is obvious that every completely
positive matrix is doubly nonnegative,
but the
converse is not always true [3, 5, 61. It depends in some sense on the zero pattern of the matrix. To describe this dependence,
we associate with an n x n symmetric
38Department of Mathematics, Technion-Israel Israel.
matrix
A a
Institute of Technology, Haifa 32000,
3QResearch supported by the C. Wellner Research Fund at the Technion.
762
FRANK
graph G defined by V(G) graph G is completely completely positive.
= { 1,
positive
UHLIG,
TIN-YAU TAM, AND DAVID
CARLSON
,n}, E(G) = {(i, j) : i # j, aij # 0). We say that a
if every doubly nonnegative
The aim of this work is to characterize
completely
matrix
positive
A with G(A) graphs.
= G is
The following
results are known:
PROPOSITION1 [5, 91.
A graph
PROPOSITION2 [2]. completely
Bipartite
graphs
(graphs
positive i$n < 5.
is completely
which
contain
no odd
cycle)
are
positive.
PROPOSITION3 [3]. If a graph then G is not completely The characterization A is completely
m-dimensional
vectors,
doubly nonnegative tive inner products, that the coordinate
G contains
an odd cycle
of length
greater
4,
than
positive. is completed
when G is not completely matrix
G with n vertices
positive. positive where
by proving
that Proposition
that an n x n
if and only if it is the Gram matrix of n nonnegative m may be greater
than
n. Using the fact that every
matrix is a Gram matrix of a set of vectors we find an m-dimensional vectors
3 is the only case
The proof is based on the observation
with mutually nonnega-
space with an orthonormal
basis such
of the set in this basis are nonnegative.
To prove our main result we use the following lemmas:
LEMMA 1. Let k be a cutpoint {k}.
If both
G, and G,
LEMMA 2.
of a graph
are completely
The graph
G, i.e.,
positive,
T, consisting
G = G, U G,
and G, fl G, =
then so is G.
of n triangles
with a common
base is completely
positive. Based on the above facts, we obtain
THEOREM 1.
Let G be a nondirected
graph
without
equivalent : (1) G is compl&ely
positive.
(2) G has no odd cycle of length greater (3) G consists
of blocks of the following
than 4. types:
(a) blocks with less than 5 vertices; (b) bipartite (c) families
blocks; of triangles
with the common
(4) All the blocks of G are completely (5) G is the root graph of a pafect
base.
positive.
line graph.
loops.
Then the following
are
AUBURN
1990 CONFERENCE
The proofs of the lemmas and of the equivalence A slightly different equivalence
proof
of the equivalence
of(l),
(2), and (3) are given in [S].
of (2) and (3) is given
in [4]. The
of (2) and (5) is given in [lo].
Let G be a nondirected completely
763
ON MATRIX THEORY
graph without loops. The set of all factorization
positive realizations
indices of
of G is denoted by Z(G). In the next theorem
we list
some facts we know about Z(G).
THEOREM 2. U G, such that G,, . . . , G, are connected by a cutpoint (see Lemma l), then Z(G) = Z(G,) + *.. +Z(G,) + {O,l}. (2) Zf K, is a complete graph, G(A) = K,, and A is a completely positive matrix, then (p(A) = rank A, and thus Z(K,) = (1, . . , n}. (3) If ) V(G)) = n Q 4 then Z(G) E {l,. . . , n}. (4) IF G is bipartite then Z(G) E { ( E(G) (; ( E(G) ( + 1). Zf G is a tree then (1) Zf G = G, U ...
Z(G) = { I E(G) I + 1). (5) ZfG=T,(seeLemma2)thenZ(G)S{n,...,2n+l}. (6) Zf there are k independent vertices in G, then Z(G) fI (1,.
. . , k} # 0.
Results (I), (2), and (5) could be improved by settling the following question:
QUESTION. Let a, b E Z(G). Does
Z(G) contain all the integers between
a and b?
The authors would like to thank Professor T. Ando for suggesting matrix theoretic proofs, using Schur complements, of Lamas Z and 2 [Z].
REFERENCES 1
T. Ando, private communication.
2
A. Berman and R. Grone, Bipartite completely
PhiZos. Sot. 103:269-276 3
A. Berman matrices,
4
at
Jerusalem,
Maximum
the
Combinatorial
k-coloring
French-Israeli
Proc. Cambridge
results
on completely
positive
(1987).
of perfect
line graphs and their roots, pre-
Symposium
on
Combinatorics
Nonnegative
factorization
and
Algorithms,
Nov. 1988.
L. J. Gray and D. G. Wilson, nonnegative
6
Hershkowitz,
Linear Algebra Appl. 95:111-125
Gavril Fanica, sented
5
and D.
positive matrices,
(1988).
matrices,
Linear Algebra AppZ. 31:119-127
M. Hall, Jr., A survey of combinatorial
of positive semidefinite (1980).
analysis, in Surueys Appl. Math. IV, Wiley,
1958, pp. 35-104. 7
M. Hall, Jr., Cumbinatorid
8
N. Kogan and A. Berman, Math., to appear.
Theory, Blaisdell, Lexington, Mass., 1967. Characterization
of completely
positive graphs, Discrete
764 9
FRANK UHLIG,
TIN-YAU TAM, AND DAVID CARLSON
J. E. Maxtleld and H. Mint, On the equation Sot. 13:125-129
10
L. E. Trotter,
RANK
Line perfect graphs,
PRESERVERS
by RAPHAEL
X’X
= A, Proc.
Edinburgh
Math.
(1962).
AND
Math. Programming
INERTIA
12:255-259
(1977).
PRESERVERS
LOEWY4’
Let V be a vector space which is one of the following: (1)
P”,
the
throughout
set of all m x n matrices
with entries
in a field
F.
We assume
that m Q n and F is infinite.
(2) H,, the set of all n x n Hermitian
matrices.
(3) S,, the set of all n x n real symmetric Given a matrix
matrices.
A, let p(A) denote the rank of A. Let
ttj=
{AEV:~(A)
For A E H,, or A E S,, let In A = (r, s,
=j).
t), where r is the number of positive eigenvalues t the number of zero eigenvalues
of A, s the number of negative eigenvalues of A, and of A. Let
C( r, s, t) = { A : In A = (r, s,
We
assume
throughout
that
k is a fixed positive
t)} .
integer
and T
: V + V a linear
transformation.
DEFINITION
(a) We say T is a rank-k preserver
if p(A) = k implies that p(T( A)) = k, i.e., if R,
is invariant under T. (b) We say T is rank-k nonincreasing
if p(A) = k implies p(T( A)) Q k. It is easy to
see that (under our assumptions) this is equivalent to the statement that the set lJf=,
Rj
is invariant under T. (c) If V = H,, or S,, we say T is a G(r, s, t)-preserver
if the set G(r, s,
t) is
invariant under T. We consider here the following three problems, which seem to have attracted of interest in recent years.
40Department of Mathemaics, Technion, Haifa 3200, Israel.
a lot
AUBURN
1990 CONFERENCE
ON MATRIX
PROBLEM
1.
When is T a rank-k nonincreasing
PROBLEM
2.
When
PROBLEM
3.
When is T a G(r, s, t)-preserver?
In thefrrst
765
THEORY map?
is T a rank-k preserver?
two problems
we assume V = Fmv”, while in
the third we assume V = H,, or
SIL.
Problem 1 A full solution functional
is known
THEOREM
1[12].
PI)
(III)
T(A)
= L’AtV
T(A)
=
(IV) It should be noted course,
nonincreasing
T is rank-l
I’( A) = UAV
(1)
depend
only for the case
k = 1. We denote
by (p(A) a linear
on A. Botta showed:
on
forsome
[a,
that in (III)
A. As indicated,
some related
THEOREM
U, VEF”‘,“,
ez(A)
a2
...
...
dA$
a,].
and (IV) ai are fixed elements
no analogous
it is clear that if T satisfies
VEF”,“,
UEF”‘,“‘,
for some
[v&4)
q and only if it is one of the following:
statement
of F and do not
holds for any arbitrary
(I) or (II), then it is rank-k nonincreasing.
k. Of
We state
results. 2 [lo]. Suppose F is algebraically
closed and T is rank-k nonincreasing.
Then either Im T c lJj”=, Rj or dim Ker T < mn - (k + 1)2. THEOREM 3 [8, 131. Suppose F is algebraically closed. If k < m and T(Ujk_ 1 Rj) c UT=, Rj, then either (I) holds, or m = n and (II) holds, where U and V are nonsingular. Note that the assumption a significant
restriction
on T in Theorem
3 means that Ker T n { UJ=, Rj} = 0,
on the rank-k nonincreasing
map T.
766
FRANK
UHLIG,
TIN-YAU TAM, AND DAVID
The following result is useful in the investigation was also used by Loewy in the investigation
THEOREM 4 [18].
CARLSON
of rank-k nonincreasing
of Problem
maps, and
3.
Suppose that T is rank-k nonincreasing.
Then it is rank-l nonin-
creasing for every 1 > k. Using this theorem,
Loewy has recently
THEOREM 5 [22]. nonincreasing,
proved the following:
Suppose that F is algebraically
closed and k < m. If T is rank-k
and Im T contains a matrix B such that p(B) 2 k + I, then either (I) or
(II) must hold. It is now clear, in light of Theorem set of rank-k nonincreasing
said if T is rank-k nonincreasing
DEFINITION.
5, that in order to completely
A subspace
It is clear that given any l-subspace For
example,
can
one
if p(A) < k for all A EL.
L, one can build rank-k nonincreasing
it is desirable
characterize
the
What can be
Q k for all A E lm T? This leads us to:
and p(A)
I_. is said to be a x-subspace
whose image is L. Therefore inclusion)?
characterize
maps we are faced with the following question:
the
to obtain information
maximal
i-subspaces
about (with
maps
Z-subspaces.
respect
to set
The task seems quite formidable.
Atkinson
and Lloyd [2, 31 and Atkinson
They defined
the concepts
weak canonical Eisenbud
of primitive
form for a %-subspace.
and Harris
roughly equivalent
[14]. They
[l] obtained
and imprimitive Another
recent
on l-subspaces.
Z-subspaces,
and obtained
paper on z-subspaces
state that the problem
to the problem
some results
of classifying torsion-free
%subspaces
of classifying
certain
the maximal
dimension
of a &subspace
of F”‘,“.
Then dim L Q kn.
a
is due to is
sheaves on projec-
tive spaces. The
problem
Flanders
showed
of determining
THEOREM 6 [15]. Flanders
Suppose L is a x-subspace
also characterized
the case where
equality
is attained.
is easier.
He assumed
that
1F 1 > k + 1, and for the case of equality also char F # 2. Meshulam [25] reproduced Flanders’s
results,
removing
the restrictions
on F. His proof uses the KGnig-Egervary
theorem. PROBLEM 2.
We assume that F is algebraically
holds, or m = n and (II) holds, where preserver. direction
The question was obtained
THEOREM 7 [24].
here is whether
closed.
It is easy to see that if (I)
U and V are nonsingular, the converse
is true.
then
T is a rank-k
The first result
in this
by Marcus and Moyls.
Suppose that T is a rank-l
preserver.
m = n and (II) holds, where U and V are nonsingular.
Then either
(I) holds, or
AUBURN
1990 CONFERENCE
Marcus
ON MATRIX
and Moyls assumed
767
THEORY
that char F = 0. Westwick
[29] obtained
the same
result for char F # 0. In their paper, Marcus and Moyls also raised the following: CONJECTURE1.
The conclusion
of Theorem
7 holds for a rank-k preserver,
where
k is any integer such that 0 < k < m. This conjecture is to date not completely resolved. We give some partial results, all confirming the conjecture. Moore [26] proved the case k = 2. Beasley [5, 7, 91 obtained various results.
They include
the confirmation
of Conjecture
1 in the cases
k = 3,
k = m, and k < $n. Recently Beasley has been able to show: THEOREM8 [ll].
Conjecture 1 h&s
solution in case F = a? It turns out that in the problem
Why do we get a complete of characterizing
rank-k preservers,
role, much as z-subspaces
a certain
family of subspaces
are associated with rank-k nonincreasing
plays an important maps.
L of Fmq” is said to be a k-subspace if p(A) = k for any
DEFINITION. A subspace
AEL,
if F = c.
AfO.
There are several papers dealing with k-subspaces.
Earlier
ones are due to West-
wick [3O] and Beasley [6]. They obtained bounds for the dimension of these subspaces. Beasley, for example, showed that if L is a k-subspace then dimL
F = @ is made. His result is somewhat too invol&l
based on Sylvester’s
work, Westwick
to be stated here.
[31] was able to show that if L is a
k-subspace of G”‘* ” then dimLgm+n-2k+l. Based on this inequality, Beasley was able to cover (for F = G) the cases k > fn that were not covered
PROBLEM 3.
in his earlier work, thus obtaining Theorem Suppose that (r, s,
t) is a fixed inertia triple, and let T : H,, + H,.
Suppose that there exists a nonsingular
(“1
8.
S E en*” and E such that either
T(A)
= ES*AS
Or
(“I)
T(A)
= ES*A’S,
where E = 1 if r # s and E = f 1 if r = s. Then T is a G(r, s, t)-preserver.
768
FRANK UHLIG,
TIN-YAU TAM, AND DAVID CARLSON
The obvious analogue holds for S,. Johnson and Pierce showed:
THEOREM 9 [17].
Suppose
that T
: H,, --) H,, is an invertibb G( r, s, t)-preserver. 0, 0), (0, n, 0), (0, 0, n), (n /2, n/2,0). Then either
Suppose that (r, s, t) is not one of (n, (V) or (VI) must hold. The corresponding
result for S, also holds. What about the four exceptional
The set G(O,O, n) consists of 0, so it is of no interest.
classes?
We clearly have G(0, n, 0) =
- G( n, 0,O). The class G( n, 0,O) consists of all n X n positive definite matrices, set of G(n, O,O)-preservers definite matrices
consists
so the
of all linear maps that map the set of positive
into itself. This is a well-known
open problem.
Pierce
and Rodman
showed:
THEOREM 10 [27]. invertible G( n/2,
Suppose
n/2,0)-preserver.
n is an even integer,
n > 4, and T
Pierce and Rodman also characterized
the set of G(l, 1, 0)-preservers,
contains the set given by (V) and (VI). The proof of Theorem proof
of Theorem
subspaces
of en,
9.
It uses the Grassmannian,
and the gap metric
the real symmetric
THEOREM 11 [20].
Suppose
such that T(A)
elements
are the
It should be noted that
n > 4, and T : S, -+ S, is an
Then there exist nonsingular
11 uses a result of Friedland
- I + 2) contains a nonzero
S E R n, n and E = f 1
9, 10, and 11 assume
Loewy and Pierce
1, any subspace
and Loewy [16] which states L of S, with dim L > i(l -
matrix whose largest eigenvalue has multiplicity
least 1. The method of proof of Theorem Theorems
whose
= ES~AS.
The proof of Theorem
dropped?
which strictly
10 is different from the
10. However, we managed to prove
n is an even integer,
that given any 1 such that 2 Q I Q n I)(2n
a space
is put into this space.
case is not covered by Theorem
invertible G( n 12, n /2,0)-preserver.
: H,, + H,, is an
Then (V) or (VI) must hoM.
11 can be used to prove Theorem that
T is invertible.
at
10 as well.
Can this assumption
be
[23] gave a positive answer for the class G(l, 1, n - 2) in
case n 2 3. Johnson and Pierce [17] gave a positive answer for the classes G(n and G(k + 1, k,O), and therefore
also for G(l, n - 1,0)
1, 1,O)
and G(k, k + 1,O). They also
stated:
CONJECTURE 2.
Suppose that n > 3 and rs > 0. If T is a G(r, s, t)-preserver,
then
either (V) or (VI) must hold. We have
THEOREM 12 [21].
Conjecture
The proof of Theorem earlier. The assumption class
G(r, 0, n - r),
2 holds ifr
12 relies heavily on rank-k nonincreasing
rs > 0 in Conjecture
where
# s.
0 < r < n. The
2 is essential. map
T
maps, discussed
Indeed, consider now the
: H, + H, defined by T(H) =
AUBURN 1990 CONFERENCE
ON MATRIX THEORY
(tr H)I,. $ 0 is easily seen to be a singular G( r, 0, n - r)-preserver. showed:
THEOREM13 [4]. Let T : H,, + H,, be a G( r, 0, n - r)-preserver, p(T) > r2. Suppose that 0 < r < n. Then (V) or (VI) must hdd.
769 Baruch and Loewy
and suppose that
It can be shown that the bound r2 cannot be improved. It comes from the possible dimensions of faces of the cone of n x n positive semidefinite matrices in H,, which are 12, 1 = O,l,. . . n. In the real symmetric case r2 should be replaced by $-(r + 1). The case r = 1 was proved earlier by Loewy [19].
REFERENCES 1
2 3 4 5 6
M. D. Atkinson, Primitive spaces of matrices of bounded rank II, J. Au&d. Math. Sot. Ser. A 34:306-315 (1983). M. D. Atkinson and S. Lloyd, Large spaces of matrices of bounded rank, Quart. J. Math. Oxford 31:253-262 (1980). M. D. Atkinson and S. Lloyd, Primitive spaces of matrices of bounded rank, J. Au&d. Math. Sot. Ser. A 30:473-482 (1981). M. Baruch and R. Loewy, in preparation. L. B. Beasley, Linear transformations on matrices: The invariance of rank k matrices, Linear Algebra Appl. 3~407-427 (1970). L. B. Beasley, Spaces of matrices of equal rank, Linear Algebra Appl. 38:227-237 (1981).
7
L. B. Beasley, Linear transformations which preserve fixed rank, Linear Algebra AppZ. 40:183-187 (1981). 8 L. B. Beasley, Linear transformations on matrices: The invariance of sets of ranks, Linear Algebra Appl. 48:25-35 (1982). 9 L. B. Beasley, Rank-k preservers and preservers of sets of ranks, Linear Algebra Appl. 55:11-17 (1983). 10 L. B. Beasley, Linear transformations preserving sets of ranks, Rocky Mountain /. Math. 13:299-307 (1983). 11 L. B. Beasley, Linear operations on matrices: The invariance of rank-k matrices, Linear Algebra Appl. 107:161-167 (1988). 12 P. Botta, Linear maps preserving, rank less than or equal to one, Linear and Mu&linear Algebra 20:197-201 (1987). 13 G. H. Chan and M. H. Lim, Linear transformations on tensor spaces, Linear and Multilinear Algebra 14:3-9 (1983). 14 D. Eisenbud and J. Harris, Vector spaces of matrices of low rank, Adu. in Math. 70:135-155 (1988). 15 H. Flanders, On spaces of linear transformation with bounded rank, J. I,.ono!on Math. Sot. 37:10-16 (1962). 16 S. Friedland and R. Loewy, Subspaces of symmetric matrices with a multiple first eigenvahie, Pa&c J. Math. 62:389-399 (1976).
770 17
FRANK UHLIG, C. R. Johnson and S. Pierce,
TIN-YAU TAM, AND DAVID CARLSON
Linear maps on Hermitian matrices: The stabilizer of
an inertia class II, Linear and M&&near 18
T. J. Laffey and R. Loewy, Linear and M&linear
19
Linear
Algebra 26:181-186
R. Loewy, Linear transformations Appl. 121:151-161
Algebra 19:21-31
transformations
(1986).
which do not increase
rank,
(1990).
which preserve or decrease rank, Linear Algebra
(1989).
20
R. Loewy, Linear maps which preserve a balanced nonsingular inertia class, Linear
21
R. Loewy,
Algebra Appl. 134:165-179 Linear
Appl. 11:107-112 22
(1990).
maps which preserve
an inertia class,
SIAM ].
Matrix Anal.
(1990).
R. Loewy, Linear mappings which are rank-k nonincreasing,
submitted for publica-
tion. 23
R. Loewy and S. Pierce,
24
M. Marcus and B. N. Moyls, Transformations Math. 9:1215-1221 (1959).
25
R. Mesbulam,
of matrices,
spaces,
Pacifi
Quart.
J.
J.
Math.
(1985).
C. F. Moore, Characterization a Tensor
on tensor product
On the maximal rank in a subspace
Oxford 36:215-229 26
unpublished.
Product
Space,
of Transformations
Univ. of British
Preserving
Columbia,
Rank Two Tensors of
Vancouver,
B.C.,
Canada,
1966. 27
S. Pierce and L. Rodman, Linear preserves balanced inertia, SZAM I.
28
J. Sylvester, conditions,
On the dimension
R. Westwick,
30
(1967). R. Westwick, 5:49-64
31
on tensor
satisfying rank
(1986). spaces,
Spaces of linear transformations
Pucifz
J.
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23:613-620
of equal rank, Linear Algebra Appl.
(1972).
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Transformations
(1988).
of spaces of linear transformations
Linear Algebra Appl. 78:1-10
29
of the class of Hermitian matrices with
Matrix Anal. Appl. 9:461-472
Spaces
of matrices
of fixed rank,
Linear
and M&linear
Algebra
(1987).
WITH
POSITIVE AND
DEFINITE
LINEAR
HERMITIAN
PART:
SYSTEMS
MATHIAS41
Let M,(C) real] matrices.
[respectively, We call A EM,
M,(R)]
denote the space of n x n complex [respectively,
positbe definite (respectively,
positioe semideftnite) if A
41Department of Mathematics, The College of William and Mary, Williamsburg, VA 23187. Research supported by an Eliezer Naddor postdoctoral fellowship in the Mathematical Sciences from the Johns Hopkins University during the year 1989-90 while the author was in residence at the Department of Computer Science at Cornell University.
AUBURN
1990
is Hermitian
CONFERENCE and
Hermitian
part
ON MATRIX
x*Ax > 0 (respectively,
and the skew-Hermitian
further
definite
results
definite
can be found
largest
Frobenius
We write
norm
use
that
A,,
being
have many
some
[respectively,
1
x
1
analogous
in this
synopsis.
and
to those Proofs
and
norm
the
alge-
. 11 2) and
(I]
semidefinite.
His proof
A,,
to denote
on M, by
Ar = b by Gaussian
is backward
being
algorithm
&,(A)]
of A. The spectral
stable
if
of A [JC~(A) = II AlI 2 II A-‘11 2 = L(
Cholesky
properties
of these
eigenvalue
solving
precise)
is not too large.
product
part
&,,,,(A)
smallest]
(I] . ]IF) are defined
to be
number
[with
The
in [6].
we
[8] showed
decomposition,
outer
x E C”.
2,
discuss
A Q B if B - A is positive
tive definite]
all nonzero
A + A*
=
Hermitian We
[respectively,
Wilkinson condition
) 0) for
of A is
matrices.
A is Hermitian,
If braically the
part
with positive
of positive
r*Ax
of A is
H(A)
Matrices
771
THEORY
was based
(n -
1)
x
elimination
A is positive A)/&,,(
(the Cholesky definite
A) because
and
on the fact that if we partition
(TV -
l)], then
we are left with the (n -
1)
after
the
A is posi-
one step
A as
of the
x (n - 1) matrix
A21 42
A=A,,-
-,
All
which
is also positive
definite
and for which
and Lax(A) which
together
imply
K~( A)
Q K~(
(2)
Q %,m,( A).
A). This fact ahm
the induction
to proceed.
772
FRANK UHLIG, When
A has positive
definite),
the leading
definite
principal
Gaussian elimination
TIN-YAU TAM, AND DAVID
Hermitian
minors
of
and run more
so one would like
efficiently,
elimination
the inequalities generalize
is not necessarily and so one
without pivoting (but there is no guarantee arithmetic).
positive definite.
(but
A are nonzero
be stable in fmite-precision Gaussian
part
without
Algorithms
to determine
will be backward
This is the motivation
for this research.
(1) and (2) to matrices argument
In [4] it was argued Gaussian elimination
with positive
heuristically
precision
definite
Hermitian
solving
Ax = b provided
that the algorithm
conditions
under
stable assuming Our approach
definite
will
structure
that
which H(A)
is
is to generalize
Hermitian
part and then
to these matrices. that
without pivoting,
in which u is machine
positive
can perform
that do not pivot preserve
pivoting
Wilkinson’s
CARLSON
i,
the solution
and c, is a linear function
part. Using this result,
to
Ax = b computed
by
( A + E) 12= b, where
satisfies
of n, when
A has positive
they argued that it is safe not to pivot when
the ratio
1)H + STH- ‘S II2
IIAll, is not large. (It is easy to show that this quantity is at least I.) We make their argument rigorous
and give a sufficient
arithmetic
(without pivoting)
completion
with positive
son’s argument, matrices First
condition
for the
pivots.
Our approach
definite
let us consider
Hermitian
some
part, in particular
the properties
a matrix.
Previous
Hermitian
part [5, 1, 2, 71 has concentrated
interlacing
inequalities
on matrices
for the arguments
this
results
follows
from
the
identity
part to run to
(Theorem
of the Hermitian
with positive
definite
part of the inverse of such
definite
on the properties
of the eigenvalues
of Wilkin2) involving
and submatrices.
of matrices
with positive
ues of AA- l* all have unit modulus.) We start by determining the Hermitian
in finite-precision
Hermitian
of inequalities
part, their inverses
of the properties
Hermitian
research
definite
is based on a generalization
and for this we need a variety
with positive
LU factorization
of a matrix with positive
Hermitian of AA-‘*,
of AA-‘*.
or skewespecially
(The eigenval-
part of the inverse of a matrix. The proof of + Y-l = X-‘(X + Y)Y-’ applied with
X -’
X = A, Y = A*. LEMMA 1. Let A have positive definite Hermitian part, and let H = H(A) and S = S(A). Then A is invertible and A-’ has positive definite Hermitian part given by
A-’
H(A-‘)
=
+
2
A-‘*
= (H + s*H-‘s)-~,
AUBURN
1990 CONFERENCE
ON MATRIX
THEORY
and we have the inequalities IIA-‘112 Define
IIH-‘II,
G
the functions
llAll,< llff+S*H-‘Sl12.
ad
j and K~ on the cone of positive
definite
matrices
by
( A-ly‘-l*)jl~H+s*H-ls
f(A)=
and K+)
where
and S = S(A).
H = H(A)
Many results matrices
involving
with positive
that K~( A) =
=
K~(
1IH+S*H-‘SI12IIH-‘ll,,
Notice
the condition deftnite just as
A-‘),
is convex with respect
that
Hermitian K~(
A) =
A) =
K~(
numbers
part when K~(
to the partial order
A) if A is positive
K~(
of positive K
definite
is replaced
matrices by
K*.
definite. hold for
Notice ako
A-‘). Th e next result, which states that f ,< , is crucial, and numerous inequalities
follow from it (we only state a few here).
2.
THEOREM
order
Q
Let
f be
defined
by (5). Then f is convex with respect
. That is, for any A,, As EM,
with positive definite Hertnitian
to the partial part and any
t E [O, 11,
f(% + (1 - +z) Furthermore, partitioned
suppose
that A, B EM,,
have positive definite Hermitian
(7) part and A is
as
A=
and bt
Q tf( 4) + (1 - t)f( As).
f( A)
with
be partitioned
A,, E Mn-k,
in the same way. Then
1.
Ijf(A22
2.
f(4,
3.
f(h)
Gf(Ah,
4.
aH(A)
2 II Allzll A-‘112
5.
q,(
6. 7.
K”(A)
- ~zlK&)((,
@
A,, E Mk,
4422) Gf(4,
~ljf(A)22112~ ef(A)zz,
= KZ(A) > 1,
tA + (1 - t) B) ~mmax{~~(A),K~(B)}f~anytE[O,1], 2 KH(&
@ AZ,) 2 KH(&)~
KH(A) 2 KH(AZZ - Az,4?‘4,).
Notice that if A has positive of A then, combining
definite Hermitian
4 and 6, we have
K~(
B) Q
part and B is a principal submatrix K~(
B) Q K”(A).
That is, we have a
774
FRANK UHLIC,
bound
on the e-norm
positive definite
condition
Hermitian
number
TIN-YAU TAM, AND DAVID
of any principal
submatrix
CARLSON
of a matrix with
part.
We also prove the following perturbation
result for the function
f by a straightfor-
ward argument.
Let A = H + S have positive definite Hermitian part, and let E be such
LEMMAS. that
[If(A) -f(A+E)II
~~ll~ll,Il~-‘IlzlI~+~*~-‘~l~~=~ll~ll~~~(~)~
Note that if we restrict stronger than (9):
A and E to be Hermitian,
IIf(A+E) -.@)]I,= regardless
of the value of
purposes,
since
A). However,
K~(
our results
in Theorem
Now we consider algorithm
without
the backward
pivoting
part using finite-precision for a complete
arithmetic.
discussion.
matrices
to Hermitian
LU factorization definite
Hermitian
a rigorous
see [3]
are real in our final result.
part of A, is the same as the Hermitian
error of the LU factorization
solution to Ax = b computed
for our matrices,
are many reasons to avoid pivoting;
already proved above and an induction
This in turn can be used to derive
is
in [S] (up to a constant).
of the outer-product
to a matrix with positive
There
[In this case, (A + AT)/2, the symmetric
R =
restricted
We will assume that all matrices
A.] Using the inequalities bound on the backward
the bound (9) is quite satisfactory
stability
when applied
which
IIEII,,
4, when
reduce to the bounds proved for Hermitian
then we have a result
(9)
argument,
part of
we have a
of A with H(A) positive definite.
bound
by Gaussian elimination
on the backward without pivoting.
error
in the
Given a matrix
[ bjj]. we define I B I = [ I bij I]. THEOREMS.
and S = S(A).
Let A E M,(R) have positive dejkzite Hermitian part, and let H = H( A)
Then L and V, the exact LU factors
IIlL Let u be machine
precision.
of A, satisfy
IUl(I,~nllH+STH-‘SII~.
lf
%n3/2~H(
A)” < 1,
(10)
AUBURN
1990 CONFERENCE
775
ON MATRIX THEORY
then the LU factorization algorithm runs to completion and the computed factors i and fi satisfy IIifi-
7un3”]]H+
All,<
S*H-lS]Js.
(12)
Block LU factorization algorithms (see, e.g., [3, Algorithms 3.2.5, 3.2.61) typically will not produce exactly the same computed LU factorization as scalar algorithms (e.g., [3, Algorithm conclusions, x2(B)
3.2.4]),
but
one
with different
may
constants
< xn( A) for any submatrix
expect
the error
in (11)
analysis to produce
and (12),
since
we have
B of a positive definite matrix
similar
shown
that
A.
REFERENCES K. Fan, On real matrices
with positive definite symmetric
M&linear
(1973).
Algebra 1: l-4
K. Fan, On strictly dissipative matrices, G. Golub and C. Van Loan, Baltimore,
component.
Linear Algebra AppZ. 9:223-241
Linear and (1974).
2nd ed., Johns Hopkins U.P.,
Matrix Computations,
1989.
G. H. Golub and C. Van Loan, Unsymmetric
positive definite linear systems, Linear
(1979).
Algebra AppZ. 28:85-97
C. R. Johnson, An inequality for matrices whose symmetric Linear Algebra AppZ. 6:13-18 R. Math&,
Matrices
with positive definite Hermitian
systems, SIAM J. Matrix Anal.
Appl., to appear.
R. C. Thompson,
matrices
11:255-269
Dissipative
part is positive definite,
(1973).
and related
part: Inequalities results,
and linear
Linear Algebra
AppZ.
(1975).
J. H. Wilkinson, International
A priori error
analysis of algebraic
Congress of Mathematicians,
WARING’S
PROBLEM
FOR
by BORIS
REICHSTEIN4’
processes,
in Proceedings
of
1968, pp. 629-640.
SMOOTH
CUBIC
Let (p be a cubic form, i.e., a homogeneous
CURVES
polynomial of degree 3. We would like
to determine the smallest integer k such that (p can be expressed as a sum of cubes of k linear forms and find all corresponding
representations
tions. This question is known as Waring’s proposed belonging
an algorithm
that allows one, for some values of k and for cubic
to a wide class of forms
42Deparhnent 20064.
that we call Waring presenta-
problem for cubic forms. In [l, 21 we have
in n variables,
to find almost
of Mathematics, The Catholic University of America,
forms
all the Waring
Washington,
D.C.
776
FRANK UHLIG,
presentations algorithm, almost
or to prove however,
all Waring
presentations
form
form
written
in canonical
linear
forms
form
appear
by cube
root
two-dimensional. investigate
In this case
coordinates in Waring
is unique
that
the cubic
presentations.
formulas
It turns
vanishes;
the variety
X,
obtained
by methods
of algebraic
that, opposite to the case of quadratic forms, X,
and
X,
work
of the we find
corresponding
(p smooth.
For the
multiplication
of Waring
inspired
geometry. is irreducible
of
k = 3 if and only if
k = 4. In the first case
of terms
in this
work
for the coefficients
out that
otherwise
(up to reordering
formulas
curve
we also call the form
of 1). In the second case the variety The
Some of the final steps
n > 3. In the present
we find explicit
[3, p. 3021 of the curve
presentation
exists.
only to the case for n = 3 provided
(p is smooth.
that
the j-invariant Waring
that no such presentation
are applicable
to the given
TIN-YAU TAM, AND DAVID CARLSON
Zinovy
the
of each
presentations
is
Reichstein
to
It has been established and irrational [4].
In order to derive the desired formulas we first exploit those steps of the algorithm in [1, 21 that are applicable to the case n = 3. The algorithm requires form (p = x: + 3x3(x:
Any smooth cubic fom
LEMMA 1. appropriate
linear
transfotmation
cp in three variables
of variables,
Let cp(zl, z2, z3) be an arbitrary
Proof. linear
cp to be in the
+ ~22) + (pp(x1, x2) where p2 is a cubic form in xl and x2.
can be written,
after an
as
cubic
form.
It is known
that there
exists a
T, : ( zlr z2, z3) + { y,, yz, y.J such that
transformation
$0( y1, yz> Y3) = Y? + Y$ + Y33+ 3OYl YzY31
(2)
0 # 0 [3, p. 2931. Let
xl
Tz: { ~1, YZ~ ~3) -+
+
ix,
J;;’
x1
J;;’
-
ix, ~3 I
Then the form Q becomes T,T,
maps
the arbitrary
as in (1) with p = 2/ 0.
form
Since the inverse transformation find all Waring presentations In order prescribes
T; ‘T;’
n
maps (1) into (p(zl, z2, .z3), it suffices to
for the form (1).
to find a Waring
introducing
Thus, the linear transformation
cp into the form (1).
presentation
the following matrices:
of the form (1) the algorithm
in [1, 21
AUBURN These
1990
matrices
CONFERENCE commute
if and
expressible
as a sum of cubes
~1 = z/2,
then
gl=
ON MATRIX only
if $
c3 = 8. Thus,
to [2], these
rp(%~,A,)
vectors
the Waring
5 j=l
: g3jkTl
gli,
g,i,.
. . , g3i are coordinates
(
-fix,+
Therefore,
for the form
form
g,=(-$-
$,lJ1.
presentation
(gljrl
+
g2jx2
+
g3jx3)3.
(4)
dj
of the vector
vGx,+2x,
3
gi, i = 1,2,3.
- fix,
-
&x2
Thus,
+ 2x3
3
+ 2% (2) with
(2) is
of Ds are
define
=
the
forms if and only if o = 0 or o3 = 8.43 If
{l,o,2E]t, g,=(-$.$Ji’.
According
where
= $. Then
of three linear
the eigenvectors
777
THEORY
I u = 2 we obtain
i the following
2%
(5) I
representation
as a sum
of three cubes:
(
(l-iv5)yl+(1+iv%)y2+2y3 2fi
i (l+ifi)yr+(l-iv5)y2+2y3 2%
13 +
1.3
430f course, this result is consistent with the fact that the j-invariant of the form (2) is a3(8 - a3)/(03 + 1)3 [3, p. 3021 and therefore the form (2) with (r = 0 is equivalent to the forms (2) with o3 = 8.
778
FRANK UHLIG,
TIN-YAU TAM, AND DAVID
CARLSON
From now on we assume
u # 0,
Under these conditions cannot be expressed
a3 - 8 # 0,
the matrices
2pa -
(3) do not commute
1 # 0.
(6)
and the form (2) as well as (1)
as a sum of cubes of three forms. In order to express
cubes of four linear forms we introduce,
in accordance
with the algorithm
c as a sum of in [l, 21, the
following matrices:
fii=
; 1P
-p
0
r
0
0IP
0
r
OS
p
Here
p, 9, r, s, and t are arbitrary
show that the commutator
0
I
El,=
)
i
complex
[Dir Da] vanishes
2~’ + r2 9=
1
P
s=
parameters.
-
p(2$
-
2,s
P(2P2
For the further
calculations
this expression the characteristic
polynomial
4(A)
=x4+
-
ci = -r{2P4
c3 = P{2Pz
-
-
1)
- 1
I)
(8)
.
for s only. After substituting
c,x3 + c2x2 + c,x + Co),
1
+ ( p2 + r2 + 3)9
-
+ 2r2
( p2 - 3r2
calculations
for
- N) of 6,:
( p” - r2 + l)$
cs = - {2P4 + (2p2
I
Straightforward
where co = - (2~” -
(7)
in (7) we obtain the following expression
4(X) = det(fir
‘i2$
t
- 3)
we shall need the expression
from (8) into the matrix 6,
r 9
- p2 + 3r2
- p2 + 3rs
rr(@
0
0 I 100’ 0 9
if and only if
’
r=
-P
-n 0 r
+ r”),
- 2( p2 - 2r2
+ 1)~~ -
+ 1)).
+ l)},
( p2 + r2 + l)},
(9)
AUBURN
1990 CONFERENCE
If h,, &, Xs, X, are eigenvalues of d,, corresponding
gi=
eigenvectors
779
ON MATRIX THEORY
they are distinct for almost all p and r. The
are
{-p&(P+X,),r(p&-
+CLjtT
l),-P(~+Ai),-A~+~(S+l)
g+
i = 1,2,3,4.
Each of the eigenvalues
Xi is a function of the parameters
(o(% X2-Tx3) =
5
:
( gljxl
+
(II)
p and r. The formula
g2jx2 +
(12)
g3jx3)3p
j= ’ g3jkGldj where
{ gii, g2i, g3i, gdi} are coordinates
find Waring presentations the coordinates
of the vector
that we have found so far, none of the coefficients
vanishes. Now we will find Waring presentations linear forms the variable the scalar product
(
x(
x3 does not appear explicitly.
of x3
that there
exists no
of x3; otherwise,
Let again p( xi, x2, x3)
a new cubic form
x2>
x3)
=
v(
x1,
x2,
x3)
-
p and r are arbitrary complex parameters,
defined later. We compute The commutator
Notice
, ) defmed in [l, 21 would be degenerate.
Xl,
of
of the form (1) such that in one of the
of (1) where more than one linear form is independent
be the form (1). Introduce
where
allows us to
of gi from (11) into (12), we obtain almost all Waring presentations
the form (1). In all Waring presentations
representation
gi (i = 1,2,3,4),
for almost all values of p and r (see [I, 21). Thus, substituting
the matrices
4(
PXl + fl2),,
(13)
and q is a function of p and r to be
(3) for the form (13) and their commutator.
vanishes if and only if
2/P4=
1
PP( P’ - 3r2)
If p and r are arbitrary parameters
’
(14
and q is as in (14), the form (13) is expressible
as a sum of cubes of three linear forms. Let s = p/r
be a nonhomogeneous
parameter.
780
FRANK UHLIG,
TIN-YAU TAM, AND DAVID CARLSON
The matrix D, associated with the form (13)-(14)
[see (3)] now becomes
_ /q ss + 3) - ss P(s2 D, =
- 3) _ p2(s2 -1)-l
- +;:31/
0
P( s2 - 3) 1
I Its characteristic
0
polynomial 4(X) = det(N
01
- Dr) is
(s2 + 1) 2p
f&(X) =x3+
-
1
x2 +
P(S2 - 3)
cl”( ss + 1) - 2( s2 - 2) x _
cl”( s2 -
ss - 3
1) -
1
(16)
q-3)
If h,, &,, A3 are the roots of the last polynomial, the linearly independent s) eigenvectors
(for almost all
of the matrix (15) are
g, = { Xi[ /.&(3 - S2)Xi + $(l
- s”) + 11,
XiS(2$
(i = 1,2,3).
(15)
.
-
l),p(3
- S2)hi + p2(1 - s”) + l}t
As above, we obtain the desired set of representations
of the form (1):
+&-3/4x,x,2=
SXl + x2) 3+
c
j=l
1
g3j
(
g12j +
) gij
+
( gljrl
+
g2jx2
+
g3jx3)3.
dj
From the results of [4] it follows that the roots of the polynomial (9)-(10) made rational by a substitution of variables
can be
p and r if and only if /.I~+ 4 = 0 and hence
u3 + 1 = 0. In this case the cubic form (8) [and therefore
(9)] is a product
of three
AUBURN
1990 CONFERENCE
ON MATRIX
THEORY
781
forms. If (r = - 1, we have
Y?+Y23+Y33-3Y1Y2Y3=
(Yl + Yz + YB)(%Yl
+ EZYZ + YB)(%Yl
+ ElYZ + Y3)p (17)
where cl and Ed are cube roots of 1:
The form (17) can be expressed
as a sum of cubes of four linear forms as follows:
It is trivial to obtain the similar formulas for the form (8) with (I = E~ and u = Ed.
REFERENCES B. Reichstein,
An algorithm
to express
a cubic form as a sum of cubes of linear
forms, in Current Trends in Matrix Theory, Proceedings of the Third Auburn Matrix
Theory Confwence, North Holland, 1987, pp. 273-283. B. Reichstein, appear.
On Waring’s
problem
for cubic
forms,
Linear
Algebra A&.,
to
782
FRANK and H. Knorrer,
UHLIG,
3
E. Brieskom
4
B. Reichstein, Z. Reichstein, On Math. -I., submitted for publication.
TIN-YAU
Plane Algebraic Waring
TAM, AND DAVID
Curoes,
Birkhsuser,
presentations
CARLSON
1986,
of plane
pp.
cubits,
1-721. Michigan
OUTPUT FEEDBACK CONTROL OF LINEAR REPETITIVE PROCESSES-A 2D POLYNOMIAL MATRIX APPROACH by
E. ROGERS44 and
Repetitive,
or multipass,
can be illustrated piece,
H. OWENS45
D.
involved
processes
by considering is processed the output,
forcing function
on, and hence
length a is constant, hence
operations
of sweeps,
or passes,
contributes
to, Yk+l(t),
processing
structural problems
and Owens
of the
over a by Y,(t),
processes
[Rogers and Smyth length
also
(1989)],
to the current M, or simply
The
termed unit-memory.
essential
studies
system. obviously appropriate
Roesser
example,
image-
model
[Rogers
Such processes and
process
are
unit-memory
can be regarded
bench-mining
systems
M > 1 passes which contribute termed
nonunit-memory
in the special
case
of
of M = 1.
as the natural generalization
results Further, totally
control
problem
Such behavior
on actual
is easily
processes.
from simulation it is clear
stability
and
Rogers
in the special from
control
[see,
for
example,
the
of its
control
44Department of Aeronautics 45SchooI of Engineering,
action in
methodology
and Astronautics,
University
studies Smyth
is required {Yk}kal.
is required, linear-dynamics
in Rogers
University
of Exeter, U.K.
and
is the
increase
possible
in amplitude and observed (1989)
contains
case of one type of bench-mining
appearing
treatment
process
which
in simulation
example,
appropriate
feature
analysis
a repetitive
generated
For
studies
that
undesirable
for
{ Yk}k 5 1 of oscillations
sequence
aspect is not necessarily feedback-based. A rigorous stability theory for the constant-pass developed
operations,
and 2D
state-space
so-called
it is the previous
pass profile.
unique
in the output
extensive
subclasses
counterpart.
from pass to pass. in field
certain
by the
exist-for
where
nonunit-memory
Hence a nonunit-memory unit-memory presence
described
0 Q t < a,
(1990)].
Repetitive directly
type
exist between
a
finite pass
acts as a forcing function on, and
0 6 t < a, k > 0, and is therefore
similarities
In
To introduce
Industrial examples include long-wall coal cutting and certain metal-rolling and strong
tool.
pass acts as a
suppose that the necessarily Y,(t)
which
or work-
of the processing
to, the next pass profile.
is one where
process
action
the material,
on the current
and denote the pass profile generated
a repetitive
contributes
by a recursive where
or pass profile, produced
[Rogers and Owens (1990)],
definition
k > 0. Then
characterized
by a series
such operations, formal
are
machining
and
Owens
to prevent
this
In particular, where
the
an latter
case has been (1990)]
using
of Southampton, U.K.
an
AUBURN 1990 CONFERENCE
ON MATRIX THEORY
783
abstract model formulated in functional-analysis terms which includes almost all known examples, or subclasses, as special cases. This has shown that two distinct concepts are required. These are termed asymptotic stability and stability along the pass; the former is a necessary condition for the latter, which is clearly required for all practical purposes. In effect, stability along the pass demands that, given well-defined inputs or driving terms, { Yk)k a r converges strongly to a steady, or limit, profile irrespective of the pass length a. The results of applying this abstract theory to a wide range of special cases have been reported [Rogers and Owens (1989a, b)], and it is known that the resulting conditions have well-defined physical interpretations. In particular, the results for the subclass of so-called differential nonunit-memory linear repetitive processes are well known and understood. This subclass includes the bench-mining systems as special cases and has the following state-space model:
y/r+l(t)= ‘X,+1(t) +
5
Djyk+l-j(t)-
j=l
Alternatively, suppose, for simplicity, that X k+r(O) = 0, k 2 0, and Y,_j(t) = 0, 0 Q t < a, 1 < j < M. Then it can be shown [Rogers and Owens (1990)] that (1) has the 2D transfer-function matrix description
Y(s, 2) = G(s, z)U(s, z), where the m
x 1 2D
transfer-function
Go(s)
(2)
matrix G(s, z) is defined by
= C(s& - A)-‘B
(4)
and Gj(s)
= C(s& - A)-‘Bj_,
+ I+,
l
(5)
FRANK UHLIG,
784 This subclass
has clear
structural
TIN-YAU TAM, AND DAVID CARLSON
similarities
to standard
or, in repetitive-systems
language, conoentional linear systems. Suppose that the previous pass terms are deleted from (l), the subscript Then
k + 1 is dropped, and the concept
the result is just the standard
(A, B, C), which is termed has transfer-function B = 0, B,_r
state-space
model
of a pass length is irrelevant. characterized
by the triple
the derioed conoentionaZ linear system in this context,
matrix
G,,(s), i.e. a constituent
= 0, Di = 0, 1 < i # j Q
and
element of G(s, z). Similarly, set
M, drop the subscripts, and ignore the concept of
a pass length. In this case the result is just the standard state-space model characterized by the quadruple
(A,
Bj_l, C,Dj),1 Q j Q M, which is termed the jth associated
conoentional linear system, and has transfer-function constituent The
structural
developing
matrix
Gj(s) of (5), i.e.,
similarities
a comprehensive
summarized control
above
theory
have
motivated
an approach
for (1) based on using directly
extending (where possible) the well-established
conventional
feasible stability tests (asymptotic
pass) based on G,(s)
Q
however,
and Gj(s), 1
to
and/or
linear systems theory.
date, this has yielded computationally remains,
another
element of G(s, z).
To
and along the
M, or their state-space realizations. There still
work to be done,
aspects of the role of G( s, z). In particular,
and this exposition
will focus on other
the following results, conjectures,
and open
questions will be addressed.
THEOREM 1.
Theprocess (1) is stable along the pass if, and only if, the characteris-
tic polynomial p(s, 2) satisfws Res>O,
P(S,2) f 0,
(.zI >l,
(6)
where
p(s,z):=
sl, - A c
-W
Q(z)
and
B(Z)
Proof.
=
5 Bj_lz-j,
Q(z)
j=l
= ,$Djz-j. 1, -
See Rogers and Owens (1991).
Suppose that (1) is embedded in an output-feedback-based unity-negaTHEOREM 2. tive-feedback control scheme dej%ed by
U(s, 2) = K(s, z)e(s,
2) = K(s,
z)[R(s, 2) - Y(s, z)],
(9)
AUBURN
1990 CONFERENCE
where R( s, z) is the reference
signal and K(s, z) has the structure
the open-loop forward-path p,( s, z) respectively.
785
ON MATRIX THEORY
and closed-loop
characteristic
of (3). Further,
polynomials
denote
by p,,(s, z) and
Then
Pc(s, z) =p(s, Z)I> Po(STz)
(10)
-
where the return difffence
matrix T(s, z) is given by
T(s,
z) = Z, + G(s,
z)K(s,
2).
(‘1) n
See Rogers and Owens (1991).
Proof.
DEFINITION. The natural definition of a pole for (1) is a pair of complex numbers (& z^)which satisfy p(s, z) = 0.
CONJECTURE. A pole of (1) has a well-defined physical interpretation used to characterize
stability in a similar manner
which can be
to its conventional
linear-system
counterpart.
OPEN QUESTIONS. 1.
What is the equivalent of the Rosenbrock
system matrix for (l), and how (if at
all) can it be used to define and answer fundamental 2.
systems-theoretic
questions?
How (if at all) can p(s, z) and T(s, z) be used in the development
controller
of efficient
design algorithms?
REFERENCES Rogers, E. and Owens, D. H. 1989a. processes, Rogers,
E. and Owens,
processes
D. H. 1989b.
with non-unit memory,
Rogers, E. and Owens, D. H. 1990. Lecture
Systems, Research E.
Stability analysis for discrete
Stability Analysis for Linear Repetitive Processes, Sci., Springer-Verlag,
D. H.
differential non-unit memory submitted for publication.
to appear.
Modeling and Simulation Studies on Bench Mining
Report, Div. of Dynamics
and Owens,
linear multipass
ZMA J. Math. Control Znfonn. 6(4):399-409.
Notes in Control and Inform.
Rogers, E. and Smyth, K. J. 1989. Rogers,
Axis positivity and the stability of linear multipass
Linear Algebra Appl. 122/123/124:779-796.
1991.
and Control,
An output feedback
linear
repetitive
processes,
Univ. of Strathclyde. based control Linear
theory
Algebra
for
Appl.,
786
FRANK UHLIG, TIN-YAU TAM, AND DAVID CARLSON
MINIMUM
PERMANENTS
ON CERTAIN
DOUBLY
by SEOK-ZUN
SONG46
I.
AND
MINIMIZING
STOCHASTIC
MATRICES
MATRICES
Introduction For a pair ( p,
q) of positive integers, J,, 9 will denote the p x 9 matrix all of whose
entries are 1. Let the set of all n-square doubly stochastic This
is known
to be a polytope
n2-dimensional
Euclidean
Let D = [d,,J
= {X=
with
n! vertices
in the
[x~,~] ~Q,(x~,~=Owheneverd,,~=O).
Euclidean
X E Q(D). Such a matrix
matrices be denoted by Q,. 1)’
matrix, and let
Q(D) is a face of the polytope
finite-dimensional
(n -
space.
be an n x n (0,l)
Q(D)
Then
of dimension
and hence,
Q,,
space, contains a matrix
being a compact
subset of a
A such that per A Q per X for all
A will be called a minimizing matrix on n(D).
Without any doubt, one of the most interesting
and important problems concerning
the face Q(D) is that of determining
the minimum value of the permanent
the set of all minimizing
on it, of which many studies have been done by
several
authors.
determined
For
matrices
example,
the minimum
Knopp
and Sinkhom
permanents
on Q( Dl),
[6], Mint
function and
[7], and Brualdi
[l]
Q( Dz), and Q( D3), respectively,
where
D,
=
_“_-~~_::-_-l-, ”
*..
D,=
1,n
1
0
0
. . .
0
1
. . .
0
1
10
. . .
1
1
. . .
1
. . .
0
1
..------
1
1’
Jn-z,n
0
1
D,=
i 1
.
1
.
1
. .:’
1
I.
Friedland [4], Hwang [5], Foregger [3], and Chang [2] also determined permanents on certain faces (see [9]).
the minimum
4sDepartment of Mathematics, Cheju National University, Cheju 690-756, Republic of Korea, and Department of Mathematics and Statistics, Utah State University, Logan, Utah 84322-3900.
AUBURN 1990 CONFERENCE
787
ON MATRIX THEORY
In [I], Brualdi determined n( W,), where
the minimum
permanent
and minimizing
1
0
0
..:
1
0
1
0
0
a**
0
1
matrix on
We wanted to extend this result from W,, to V,,,,nr defined as
In [ll], we determined the’ minimum permanent and minimizing matrices on Q(V,,J for arbitrary n. In [lo], we determined them on Q(V,,s) for m = 2 and m 3 5, but not for m = 3,4. In [12], we determined them on Q(V,,,) for all m 2 3 by a method somewhat different from that in [lo]. In [12], we also determined them for W,,” and
W,, .(O), where I
02,”
I m,m 1 I 12,”
WIn.” =
----L_-__
i
1 7l.m ,’
I
1 ’
An-2.m -----
%“(q
I Om-2,n 1------
02 _--F_~___~“_
=
[
I n, m
I
12 I
1
for n ) 2, m ) 3. LEMMA 1 (Foregger
[3]).
Let D = [d, j] be an n x n fdy
indecomposable
matrix, and A = [ ai, j] be a minimizing m&-ix on Il( D). Then A is fdly
(0,l)
indecomposable,
and for (i, j) such that di, j = 1,
per A( i 1j) = per A
if
ai, j > 0,
per A( i 1j) > per A
if
ai,j = 0.
LEMMA 2 (Mint [7J). [d,, . . . >d,],
and if&
If A = [al,. . . , a,] is a minimizing matrix on Q(D), D =
= d2, then
per[cual+/3a2,pa,+ora2,a3,...,a,] fOranyol,@>Owithcu+j3=1.
=perA
788
FRANK UHLIG, TIN-YAU TAM, AND DAVID CARLSON We will call this the aweraging method when c~ = /_?= i.
THEOREM 1. For m > 3, let
Then a minimizing matrix form A on Q( V,,,,3) is
(I.‘) and the minimum permanent is mlam-‘[(m
- l)mb4 + 2maxb2+
P-2)
?a”],
where mu = 1 - 3 b, x = 1 - mb, and b is a real root of 27 + m
m2+ 6m+20
21 - ?. m
9 3 +m” m3
b4+
i
b-$=0.
(1.3)
THEOREM 2. For m p 2, n 2 3, let
I Om-2,n
Wm,Il=
/ m,m 1
I --- L__*Ln__ 2, i J“,rn ,’
Then the minimum permanent un O(W,,,,) m! &c-
1 .
is
4(n - l)(n g+l
- Z),-’
(2.1)
AUBURN 1990 CONFERENCE
ON MATRIX THEORY
789
fwna4,and
(m - l)!
5b + 6mb2
2mmm2
fm n
(2.2)
= 3, where b is the unique real root of the equutiun
llm2b3
- 16mb’ + 9b - 2 = 0. m
(2.3)
We remark that the matrix W,,” is cohesive and not barycentric for m 2 2 and n ) 4. (For definitions, see [l].) By the averaging method of Lemma 2 and the proof of Theorem 2, we obtain the following result about one of the faces of n( W,. J: COROLLARY3.
For m 2 2, n > 3, let
Then the minimum permunent on Q( W,,,, *(O)) is the same as (2.1) in Theorem 2, which occurs at the barycenter b(W,,,, JO)), where the batycenter of Q(D) is giuen by b(D) = P, and the summation extends over the set of all permutation matrices P (llP~D)&<, with P < D, and per D is their number. REFERENCES R. A. Brualdi, An interesting face of the polytope of doubly stochastic matrices, Linear and Multilinear Algebra 17:5-18 (1985). D. K. Chang, Minimum permanents of doubly stochastic matrices with one fixed entry, Linear and M&linear Algebra 15:313-317 (1984). T. H. Foregger, On the minimum value of the permanent of a nearly decomposable doubly stochastic matrix, Linear Algebra Appl. 32:75-85 (1980). S. Friedland, A proof of a generalized van der Waerden conjecture on permanents, Linear and Multilinear Algebra 11:107-120 (1982). S. G. Hwang, Minimum permanent on faces of staircase type of the polytope of doubly stochastic matrices, Linear and M&linear Algebra 18:271-306 (1985). P. Knopp and R. Sinkhorn, Minimum permanents of doubly stochastic matrices with at least one zero entry, Linear and Multilinear Algebra 11:351-355 (1982). H. Mint, Minimum permanents of doubly stochastic matrices with prescribed zero entries, Linear and Multilinear Algebra 15:225-243 (1984). H. Mint, Permunents, Encyclopedia Appl. 6, Addison-Wesley, 1978.
FRANK UHLIG,
790
TIN-YAU TAM, AND DAVID CARLSON
9
H. Mint, Theory of permanents
1982-1985,
10
(1987). S. Song, Minimum permanents
on certain faces of matrices containing an identity
submatrix, 11
Linear Algebra Appl. 108:263-280
S. Song,
Minimum
permanents
Algebra AppZ. 143:49-56 12
Linear Multilinear Algebra 21:109-148
(1988).
on certain
doubly
stochastic
matrices,
Linear
(1991).
S. Song, Minimum permanents
on certain
doubly stochastic
matrices
II, Linear
Algebra Appl., to appear.
ON RATIONAL by EIVIND
1.
n
MATRICES
and BOONCHAI
K. STENSHOLT4’
and Observations
Let n, t, x be integers,
1.
DEFINITION x
STOCHASTIC
STENSHOLT4’
De$nitions
set of n
DOUBLY
matrices
with integer
entries
n > 0, t > 0, r ) 0, and N,(t, x) be the from
(0, 1, . . . , t} such that all row and
column sums equal r. Let M,,(t) be given by
iv,(t)
The cardinahty
A
of a set S is denoted
COMBINATOBIAL
owners
hold
u
= N”&O)
Aqt,1)
each,
none
0..
u N&q.
) S (.
In each
INTERPRETATION.
r shares
u
more
than
of n companies
t shares
are
x shares;
in any company.
n
A matrix
(aij) E N,(t, x) specifies a possible distribution of shares, owner i holding aij shares in company j. The special problem Anand, matrices
Dumir,
of determining
and Gupta
[l];
in NJ r, r) are sometimes
indicates additional requirements the entry set is {1,2,.
47Norwegian School
( NJ r, r) ] was studied by Mano [14] and
they denote
] NJ X, r) ( as H(n, r)
called magic squares,
for r = x. The
although usually this term
(such as that the two diagonal sums also equal r and
. , n2}; see [2]).
of Economics
481nstitute of Marine Research,
P.O.
and Business Administration, 5035 Bergen, Box 1870,
5024 Bergen,
Norway.
Norway.
AUBURN 1990 CONFERENCE Several properties
791
ON MATRIX THEORY
of these matrix sets follow from the definition:
N,( t, x) = 0
if
t
(1.1)
N”( t - I, x) c N”( t, x)
if
m -‘
(1.2)
N,( t - 1, x) = N,( t, x)
if
x+l
(1.3)
IN,(t,x)l=IN,(t,nt-x)1. [For (1.4) consider
(1.4)
the l-l map (ajj) * (bij) where aij + bij = t for all i,j.]
NOTATION. Let Q, be the set of doubly stochastic n x n matrices 31, and aQ, its interior and boundary in the relative topology of its &ne span W of dimension m = (n - 1)‘. Let r,(x) be the lattice of n x n matrices with entries zx-I, z E Z, and C,,(e) the n2-dimensional cube of n x n matrices with entries in [O, e]. J, denotes the n x n matrix where all entries equal 1. The set 0, is the intersection of W and the nonnegative orthant of @“‘. Its geometry was studied in [4]. From Definition 1 it follows that for x 2 1,
AEN,,(t,
x)
ifandonlyif
n n,.
n C”(K’)
x-~AEI’,(x)
(1.5)
By (1.5) NJ t, r) splits up as follows if x >, 1: N,(t,
TX)= I$( t, x) U aN,( t, x) (disjoint union),
(1.6)
where &(t,
x) = {AEN,@,
and
x)(x-‘Ad,}
aN,(t,
x) = {AEN+,
x)/x-‘A&Q,).
Let (aij) E NJ t, r), x > 1. a62, consists of those matrices in Q, which are limits for convergent sequences in W \ Q,; hence (aij)
Efin(t,X)
(aij)EaN”(t,
X)
ifand only if
aij > 0
for all i, j,
(1.7)
ifandonlyif
aij=
forsome
(I .8)
0
i, j.
From this it follows that
qt,
x) = 0
&(t* n) = {k) AsN,(t
-
1, x - n) ifandonlyif
if
l
when A +],,Efi,,(t,
(1.9) (1.10)
t>l,
x)
when
x > n.
(1.11)
FRANK
792 We notice the following
UHLIG,
consequence
TIN-YAU TAM, AND DAVID
of these observations:
IN,(x-n,x-n)I=IN,(x-1,x-n)I
and
CARLSON
By (1.3) and (1.11)
IN,(x-1,x-n)(=l~“(x,x)(; (1.12a)
hence
p,(x-n,r-n)l=p”(x,x)I, 2.
Triangulations
Proper
A triangulation of any polytope 62 is a finite set y= {T,, Ts, . . . , T,} all faces of simplices in ? are in 9-, n = T, U T, U *. . U T,, and
DEFINITION2. of simplices
where
if Tj tl Tk # 0,
moreover,
(1.12b)
x>n.
said to be proper
then Tj 17 Tk is a face of both Tj and Tk. The triangulation
if all corners
of each Tj belong to the corner
is
set of fl.
THEOREM(Fuglede [8], Brbndsted [S]). Let Q be a d-dimensional conuex polytope with corners p,, p,, . . . , p,. Then there exists a proper triangulation of Cl. Birkhoffs matrices
permutation taining r,(x)
theorem
for corners. matrices
x- ’ * ( aij).
[3] states
that
For a proper
51, is a convex
triangulation,
that are comers
in the unique
( N,,(cc,x) 1 is determined
polytope
(aid) E
with the permutation
N,(t, x) is a unique sum of the
lowest-dimensional
by counting
the number
simplex
con-
of points from
in each simplex. So, let A be a simplex
with the permutation
matrices
PO,P,, . . . , Pd for comers.
The points
PO+ CkiX-‘(Pi form a sublattice A of r,(x).
- PO),
kiEZ7
l
If d Q 3, it is easy to see that A = r,,(x) whether
A actually
(2.1) fl D, D being the
alline
span of A. It is not known
can be a proper
r,(x)
tl D for (high) values of d, but the possibility must be considered.
sublattice
of
Since the two
lattices have the same affine span, A is of finite index, say s, in I’,,(x) il D. From the kth coset we pick the unique representative
Qke
{p.
+
=j+x-‘(Pi
Qk such that
- P,,)jO < ri G 1, 1 G i 6 +
thus r,(x)nD=(~,+n)~(~,+a)u~~~u(~,+A).
(2.2)
AUBURN Let
1990 CONFERENCE
Qi + A = A, i.e.
793
ON MATRIX THEORY
Qi = x-‘[(x
- d)Pa + P, + .*.
+Z’d]. Each
x-l*
(aij)~r,(x)
fl n, is counted in the simplex of smallest dimension which contains the point, i.e., we count the inner lattice points in each simplex. Using
we count layer by layer. Here
y = z ‘when we count the points of (Qr + A) tl A; the
points is in a face of A and is not counted.
Qk = Pa + -&ix-1(P,
- P,,),
Now, write
0 < q.‘kiQ 1,
(2.3)
and let bi E Z be such that b, Q Crki < b, + 1. Then, for 1 < k < s, we have 0 < b, < b, = d, and the number of points from Qk + A in the interior H of A is
(2.4) thus (r,(x)nAI=
2
‘+d,‘-,,i,
for k>l.
d=b,>b,
(2.5)
k=l
Stanley [17] has proved the following result, originally conjectured
in [l]:
THEOREM (i) ] N,( x, r) ] is a poZytaomiaZin r of degree (n -
l)‘,
x )
1.
When x is allowed to assume also nonpositive values, (ii) the polynomial has n - 1 zeros: IN”(_l,
-
l)/
=
**.
=IN,(l
- n,l
- n)I = 0,
and (iii) it has the following symmetry
1%(-n-x9
-
A simple proof of (i) and (ii).
property:
n-x)/ = (-l)“-‘(~,(~, The exi$ence
of proper
x)I, triangulations
and (2.5)
show that ]N,(r,x)l, la&(x,x)], and I&,(x,x)] = INn(x,x)) - l?JN,,(x,~)l all are polynomials for x > 0. So (i) holds, and (ii) follows from (1.9) and (1.12) with x=1,2 ,..., n-l. n
FRANK UHLIG,
794 (iii) is a special
case of Ehrhart’s
TIN-YAU TAM, AND DAVID
reciprocity
deeper nature than (i) and (ii); see Remark
theorem
CARLSON
[7]. It still seems to be of a
2. A proof in line with this account
is in [12];
see also [6, 7, 15, 171.
REMARK 1. If s = 1 for a simplex of maximal dimension s = 1 for each
of its face simplices
too.
Since
tributed,
the volume of a maximal-dimensional
construct
maximal-dimensional
such that the number increasing integers;
i. [Then hence
simplices
of zero
(n -
points
1)2, then clearly are uniformly
simplex is proportional
in
P, + P, + * . . + Pi decreases
PO, P,, strictly
fl W and x:zi = 1 imply that the
X.ziPj = (aij) Ed,
dis-
to s. It is easy to
such that s = 1 in (2.5); just choose
entries
. with
xzi are
( uij) E A.]
PROBLEM. Are there maximal-dimensional are there
the lattice
cases with
s > l? If so, are there
simplices proper
with different
triangulations
volumes,
which
i.e.,
avoid these
cases? Let the polynomial
1N,,(x, x) ) be expressed as follows:
(2.6) A proper
triangulation
simplices.
The symmetry
a small number computer. With
The
more
functions
s = 1 for all simplices
(iii) allows the polynomials
of values; sequences
refined
completed
with
for
n < 5 sufficiently
{ ( N,( x, x) ( },
techniques
must
many values
r = 0, 1,2,
and a computer, a,
a,_i
ai i-dimensional
have been
from
found
by
. . appear in [16] for n = 3,4.
Jackson
the case n = 6 too. They also gave equivalent [17]. Thus one obtains
contain
) NJ x, x) 1 to be determined
and van Rees
[lo]
have
results in terms of generating
a, for n = 3;4;5:
.
3 12 19 15 6; 352 2464 7544 4.718075
2,905,658,575 14,062,951 These
13,232
51.898825
14,620
2,463,775,850 1,784,345
coefficients
indicate
complex.
In a proper
with
common);
9 vertices
1468 258 24; 1706.729525
2584.561500
770,476,155
280,134,105
74,580,465
120. triangulations,
ai > 0 and u,_r
with ai i-dimensional
= [n + $(n” - 3n + 2)]a,.
simplices, So the sim-
seem, on average, to have n facets (faces of dimension remaining n2 - 3n + 2 are walls in the triangulation
triangulation of valency
of Qs, the 3-dimensional
simplices
4 (adjacency
a 2-dimensional
is to have
similarly Iah+
indicates
6090
that proper
plices of maximal dimension m - 1) on aa,,while the
4945
811.572625 1,584,408,615
140,740
exist and are very regular:
graph
10,532
262.803150
x)1 =IN&,
a graph with 1408 vertices
x)] -IN+
- 4, x - 4))
of valency 9 on aQ,.
on an,
form a face
in
AUBURN 1990 CONFERENCE
795
ON MATRIX THEORY
PROBLEM. What is true for general
n?
Factorized forms of (2.6) for n = 3,4 illustrate (ii): 8- ‘( x + I)( x + 2)[( x + l)( x + 2) + 21 and 11,340-l( x + l)( x + 2)(x + 3)[11( x + 2)6 + 23(x + 2)4 + 128(x + 2)’ + 3061. The first is (33) in [l], and MacMahon [13, Section 4071 gave the form
3(x:3)
REMARK2.
+ (x:2).
By (1.6), (1.12), and part (iii) of Stanley’s theorem
P”(X, x)1=lN,(--)I -I NJ r -
we have
.,x-.)I=IN,(x,x)l+(-l)“lN,(-x,-x)1;
hence (aN,(O,O)j
= [1 + (-1)“]
*IN,,(O,O)l.
(2.7)
For x = 0, (2.5) becomes
i
-;I=(_l)d
Summing for the simplices in 62, and an,, we see that ] N,,(O,0) ] and ] aNJO, 0) ] are the Euler characteristics of 0, and do,, i.e. of the ball and sphere of dimensions m and m - 1. Hence (2.7) is a topological relation which follows from (iii); for this reason it would be interesting to prove (iii) as simply as (i) and (ii). [From topology ] N,(O, 0) ] = 1, i.e., the polynomial gives the correct value also for x = 0.1
3.
Error-Correcting
Codes Related to fl,
The set M,,(t) from Definition 1 is a code with alphabet {0, 1,2, . . . , t}. If (aij) E M,(t), n >, 3, is received with an error in a single entry, this is located by means of the deviating row and column sums, and uniquely corrected. The Hamming distance is at least 3 if n = 3, at least 4 if n is at least 4. This code may well be used as an identification code like the ISBN book code [ll], the universal product bar code UPC [18], or the codes for bank account numbers. For general information see [9]. These well-known codes detect errors only, and a code which also corrects errors may be a worthwhile alternative. The t&,-codes proposed here can be organized in different ways, e.g. according to the row and column sum x, and according to the simplex in a given proper triangulation to which a codeword is associated. Thus codewords may be assigned systematically and without repetitions, e.g. by different local authorities, each with its own simplices. The connection of these codes to the polytopes II, is described in (1.5). We report a few formulas for ( N,( t, x) I and ] M,,(t) ] = C ] N,,(t, x) ( . Apart from the case n = 2,
FRANK UHLIG, TIN-YAU TAM, AND DAVID CARLSON
796
where one may list all codewords,
lM4k
- l)l=
they are based on computer
5578~”
&
+ 12,705~’
+22,520n4
JN2(u-l,x)l=u-
-18(u2
1)2 -
Difference
- 2)1=
+ 1008~’
--&u(1007u8
schemes show that
12(u2+
+ 1)(2x
if u - 1 ,< x < 2u - 2 [otherwise use (1.4),
- 1,2u
+ 19,614u6
+ 14,175
jr-u+ll,
INa(u - 1, x)1 = &[21(u2+
IA?+
counting of the N,( t, x):
1) + 4
- 3u + 3)” +5(2x
- 3u + 3)‘]
(1.3) and reduce to x = t = u -
+ 2766~~
+ 3759u4
+ 3844~~
11,
- 36).
1N4(t, x) 1 fits no polynomial formula, even for t < x <
3t.
PROBLEM.
Get formulas or other results for
1M,(t) 1 directly.
Part of the work was done while the authors were visiting at the University of Wisconsin - Madison with support from the research funds at the Norwegian School of Economics and Business Administration and the Norwegian Council for Science and Humanities (NAVF). The authors are grateful to these institutions, and to Hedge Tverberg fm essential references. REFERENCES 1
H. Anand, V. C. Dumir,
and H. Cupta,
2
(1966). W. H. Benson and 0. Jacoby, New Recreations with Magic Squares, Dover, 1976.
A combinatorial
distribution
problem,
Duke Math. J. 33:757-769
3
4
G. Birkhoff, Tres observaciones sobre el algebra lineal, Rev. Uniu. Nat. Tucuman Ser. A 5:147-151 (1946). FL A. Brualdi and P. M. Gibson, Convex polyhedra of doubly stochastic matrices, I. Applications of the permanent function, /. Combin. Theory A 22:194-230 (1977); II. The graph of Q,, J. Combin. Theory B 22:175-198 (1977); III. Affine and combinatorial properties of n,, J. Combin. Theory A 22:338-351 (1977).
AUBURN
1990 CONFERENCE
ON MATRIX THEORY
5
A. Brbndsted,
Continuous
barycenter
6
Math. 4:179-187 (1986). W. Dahmen and C. A. Micchelli,
functions
797
on convex pOlytOPes, Exposition.
The number of solutions to linear diophantine
equations and multivariate splines, Trans. Amer. Math. SOC.308509-532 7
E. Ehrhart, DBmonstration de la loi de &ciprocite Acud. Sci. Paris 265A:91-94 (1967).
8
B. Fuglede,
Continuous
Math. 4:163-178 9
D. M. Jackson
11
G. Knill, International I. G. Macdonald,
integer matrices,
SIAM].
standard book numbers,
Polynomials
Math. Sot. (2) 4:181-192
in the marketplace,
Amer.
Mbh.
associated
of generalized
double
Comput. 4:474-477 (1975). Math. Teacher 74:47-48 (1981).
with finite cell-complexes,
1.
London
(1971).
Combinatoq
Analysis I, ZZ, Cambridge
in one volume by bhelsea, 14
Modular arithmetic
and G. H. J. van Rees, The enumeration
nonnegative
P. MacMahon,
C. R.
(1988).
12 13
(1988).
selection in a convexity theorem of Minkowski, Exposition.
J. A. Gallian and S. Winters,
stochastic
rationnel,
(1986).
Monthly 95:548-551 10
du polyedre
U.P.,
1915,
1916; reprinted
1960.
K. Mano, On the formula of ,,H,,
Sci. Rep. Fuc. Lit. Sci. Hirosaki Unio. 8:58-60
(1961). 15
P. McMullen,
Valuations
and Euler-twe
relations
for certain
classes of convex
polytopes, 17
Proc. London Math. Sot. 35:113-135 (1977). N. J. A. Sloane, A Handbook oflnteger Sequences, Academic, New York, 1973. R. P. Stanley, Combinatorics and Commutative Algebra, Prog. Math. 41, Birkhlser,
18
E.
16
1983. F. Wood,
Self-checking (1987).
codes-an
application
of modular
Teacher 80:312-316
Received 19 August 1991; final manuscript accepted 30 August 1991
arithmetic,
Math.
Directions
in Matrix Theory, Auburn 1990, Conference
Report
Frank Uhlig and Tin-Yau Tam Mathematics Department Auburn University Auburn, Alabama 36849-5307 and David Carlson Mathematical Sciences Department San Diego State University San Diego, Calgornia
92182
Submitted by Richard A. Brualdi
1. INTRODUCTION
The
idea for the
conceived There,
a large number
and common however,
fourth
linear
algebra
conference
of researchers
language
that as wonderful
as these
was clearly
conferences
are,
or the direction
as discontinuous
apparent,
they give only a momentary But
will not yield the derivative f
And a direction
as research
over
that our area is experiencing.
of many discrete values off
of f. One has to work harder.
assess for something
was
and the vitality
It became
As one attends such conferences
the years, one might get a feeling for the development in calculus-knowledge
evident.
University
(see LAA, Vol. 121).
from many lands had gathered,
of our research
glimpse of the state of the art in linear algebra. -as
at Auburn
by Frank Uhlig during the 1987 Valencia Conference
developments
is especially difficult to in a vast area such as
linear algebra. So the idea was conceived
of inviting many experts in linear algebra to a conference
and asking them to present their visions of the forces from the past and those leading into the future as regards linear algebra. Frank talked with several of the participants
in
Valencia and wrote many letters from Coimbra in the fall in order to plant the seeds for this conference. “Hilbert’s
The responses were encouraging.
shoes”
would
speaker speak from his/her research:
“What
not fit anyone.
There was a general realization that
But Frank
kept on, suggesting
that every
own standpoint and develop the vision that we need for our
are the important problems?”
LINEAR ALGEBRA AND ITS APPLICATIONS
“Why are they important?’ 162-164:711-797
“Where
(1992)
0 Elsevier Science Publishing Co., Inc., 1992 655 Avenue of the Americas, New York, NY 10010
0024~3795/92/$5.00
did 711
712
FRANK
UHLIG,
TIN-YAU
TAM, AND DAVID
CARLSON
they come from?’
“Where might they lead?” “ What is linear algebra as a discipline capable of solving now or in the near future?” “What lies ahead?” “How are past results pointing us in certain directions?” “ What direction is matrix theory taking?’ Thus the title for our conference. number
In this Special
Issue you will find an unparalleled
of deeply visionary papers on the affairs of linear algebra.
a year after the conference,
it is apparent
that “Hilbert’s
And now, more than
shoes” are not really needed.
We can each wear our own. One important ence.
The
Meetings
new direction
Undergraduate in Louisville
Undergraduate renewed
Algebra
in January
1990,
Conferences
in the teaching
Thompson
came
lectures
conference
aboard
and feature
all active
much needed
Together
research
of 1988.
With him he brought
in TEX and word processing. There
were
33 invited
given, with a total attendance From the beginning to support
of linear
hoped-for camellias
support
in lieu of speakers’
Our financial Auburn The
faltered, support
Uniuersity
We
specifically
Auburn University
to give
in the fall
speakers
for the success
65 contributed
of the
talks were
from 25 countries.
to use the funds that might become
according
our senior
and others
and helped us greatly with his
In addition,
of 135 matricians
speakers
algebra.
Tam joined
speakers.
the
the program of the
Tin-Yau
talents
a
issues.
during
activities,
we had planned
our invited
of 1988
on regional
Most of all we have to thank our group of invited conference.
also with teaching
we tried to balance
his extraordinary
on the
Matrix Theory
to report
surveys of some areas.
expertise
that future
in the summer
areas
at the Joint
in August 1990, represent
will involve themselves
at Johns Hopkins.
at our confer-
Discussion
and Mary Workshop
We expect
as a coorganizer
wanted to offer some minisymposia
Panel
in Williamsburg
of our subject.
at Auburn and elsewhere Carlson
too late to be represented Curriculum
and the William
Linear Algebra Curriculum
interest
Dave
appeared
Linear
to their
speakers
need.
came
And when
through
one
available agency’s
for us by accepting
fees. What a treat it is to work with matricians!
came from four sources:
and its Vice-President
Oak Ridge Associated
of Research (ORAU)
Unioersities
supported
supported
us generously.
us with
unrestricted
funds, a fact that was very helpful in the initial stages when we had to build our logistics base here and had to print and mail out the announcements
etc. in order to even have
the conference. The
National Security Agency
supplied
adequate
funds for our speakers’
expenses
where needed. Finally we are grateful to the Linear Algebra Group of SlAM and to ZLAS, who each helped us out with mailing labels and general And, last but not least, the participants Our sincere
thanks to all that were involved for making this conference
Let us conclude linear algebra and
Linear
Figure
support. who came to attend the show.
our preface
to the conference
over the last 20 years. and Multilinear
Algebra
1 shows graphs of the bookshelf
year intervals,
and with
the number
The journals were
report
Linear Algebra
founded
in 1968
space occupied of volumes
with a factual
possible. assessment
of
and Its Applications
and 1973
respectively.
by each marked on the left in 5
per period
marked
on the
right.
AUBURN
1990 CONFERENCE
ON MATRIX
713
THEORY
50
100 90
Shelfspace in cm
80
40
60 -
30
Number
50 40
-
30
-
of
volumes 20
10
20 10-
1
O-
1968 1972
1982
1 977
0
19t
(a)
30
-
25
-
- 15
10
20 -
Shelfspace in cm
15
-
-
Number of volumes
t
10 5 1 OJ
1973
31977
1982
1987
‘90
(b) FIG.
Cumulative
1.
Linear
and Multilinear
These
graphs
seem
the two journals numbers
totals
by period:
Algebra,
1973-1990.
to suggest
doubles
of participants,
that
about every
These progressions
A doubling
the
This is corroborated
1968-1990;
of papers
in each
and talks given at the Auburn growth pattern
for a constant
a = (In2)/10
of
Conferences
in linear algebra:
. volume of papers(year
(b)
by data on the
2 below.
t) = (1 + tr)
every 10 years is achieved
volume
Appl.,
10 years.
suggest an exponential
volume of papers(year
Algebra
currently
home countries,
from 1970 to 1990, as shown in Figure
(a) Linear
(t = 0.07.
1)).
FRANK
150
UHLIG,
TIN-YAU
TAM, AND DAVID
CARLSON
100I-
25
90I120
80
20
90
60
15
a
lo
ii :: t
50 60
40
r
30 30
20
0
1970
Note that Bob Thompson, an exponential
was obtained
19’80
Auburn Conferences,
in the introduction
19’S;
0
19;o
1970-1990.
to his address for this conference
growth rate for linear algebra with d = 0.04.
by counting
s
&‘--
FIG. 2.
established
::
5
A
__--
10
0
rf IFi
70
the number
of reviews
in the 15Xxx section
also
His rate figure in MathernaticaZ
Reoiews each year from 1940 on. It stands to reason that, given an overall yearly growth of 4%, a much larger growth rate of 7% is obtained Linear
Algebra,
as these were founded
only recently,
The main part of this report contains abstracts
for the two journals
specializing
in
in 1968 and 1973.
the following:
of invited talks, titles of contributed
talks, and synopsis of both.
Of course, we do not repeat those abstracts
or titles of talks that evolved into papers
in this special issue.
2. ABSTRACTS Generalized
OF
INVITED
TALKS’
inverse Invariances
by Jerry K. Baksalary.2 A survey is given of criteria
for the concepts
singular values, etc. to be invariant
with respect
such as range, rank, trace, eigenvalues, to the choice
of a generalized
inverse
‘Only those abstracts are given here that did not result in a paper in this issue. “Department PL-65-069 SF-33101
Zielona Tampere
of Mathematics G&a,
Poland,
10, Finland.
and Statistics,
Tadeusz
and Department
Kotarbiliski
of Mathematics,
Pedagogical University
University, of Tampere,
AUBURN
1990
B-,
they
when
type
CONFERENCE are referred
are relevant
estimability,
ON MATRIX
to in the product
to several
statistical
characterizations
and properties
AB-C.
problems,
of the
of canonical
715
THEORY The invariance
for instance
minimum-dispersion
correlations
in linear
properties
to criteria
linear
of this
for unbiased
unbiased
estimators,
models.
LAP&X-a Portable High-Performance Linear-Algebra Library by Javes Demmel.3 The goal of the LAPACK project library
for efficient
based
on the
eigenvalue including
used
problems,
and
recent
SVD algorithm tion ric
in the library.
First,
and the grading which
thought
positive
contain
definite
small
significantly
singular
improve
using
criterion) and
SVD.
we
show
that
work
with
overview
errors.
Third, more
In fact, even
much
project,
algorithms which
respects
a new bidiagonal
more
accurately
than
differential
we show accurate
as long
using
equa-
that
Jacobi’s
for the symmet-
as the
infinite
is
solving,
of the
solver
a Hamiltonian
is uniformly
library
high-accuracy
we present
and vectors
of roundoff
stopping
a brief on new
Second,
involves
linear-algebra The
for linear-equation
a linear-equation
values
The proof
errors,
After
we discuss
eigenproblem
relative
squares.
we concentrate
propagation
(with a modified
a portable computers.
EISPACK packages
of the problem.
computes
possible. the
and
least
results,
to understand
method
linear
and implement
of high-performance
LINPACK
benchmark
the sparsity
previously
is to design
on a variety
widely
to be included both
use
matrix
precision
entries will
not
on Jacobi.
This talk represents Deift,
J. Dongarra,
L.-C.
Li, A. McKenney,
joint
J. Du
Croz,
E. Anderson,
I. Duff,
D. Sorensen,
M. Arioli,
A. Greenbaum,
C. Tomei,
Z. Bai, J. Barlow,
S. Hammarling,
W.
P.
Kahan,
and K. Veselic.
Structured Linear Algebra Problems in Signal Processing and Control4 by Paul van Dooren. We give a survey processing though
and
the problems
no longer
make
3Department Current address: 94720.
of a number
control, use
where
one wants here
of linear-algebra the
structure
to solve for these
of standard
problems
of the
matrices
linear-algebra
occurring
matrices tools,
involved
are rather since
the
in digital
signal
is crucial. classical, structure
Al-
one can of the
of Mathematics, Courant Institute, 251 Mercer St., New York, NY 10012. Division of Computer Science, University of California, Berkeley, CA
4This paper appeared in Numerical Lirwar Algebra, Digital Signal Processing and Pam&l Algorithms (G. Golub and P. van Dooren, Eds.), NATO ASI Ser. 70, 1991, pp. 361-384. 5Philips Research Laboratory, Ave. Albert Einstein, gium; current address: Coordinated Science Laboratory, Champaign, Urbana, IL 61801.
4, B-1348 Louvain-la-Neuve, BelUniversity of Illinois at Urbana-
716
FRANK
matrices
UHLIG,
has to be taken into account.
how structure
affects the sensitivity
TIN-YAU
We discuss
Structured
matrices
around
for a long time
algebra
several
fields.
dealing
with
matrices,
structure
such
of the matrices
but one is then
from loss of accuracy divergence
of the algorithm
The importance gebra problems recognized occur.
When
them.
We
sensitivity
there
then
are fast algorithms
analyze
of a problem
if the
may have
exploited
in general.
LAPACK:
A Linear-Algebra
by Jeremy
on the
is planned
squares problems, The library
systems
common linear-algebra This library,
packages
squares.
LINPACK
These algorithms.
problems
and how
EISPACK
architectures
talks describe routines,
they
discuss
may affect
structure
the that
could
be
called
LAPACK,
for the analysis
equations,
linear
and least-
problems. a uniform
set of subroutines
solving,
networks,
among machines tools
scheme
not only will ease
of different
for evaluating
and widely used
eigenvalue an important
but they were not designed now becoming
to solve the most
on a wide range of architectures.
via computer
have provided
notes
package
algebraic
also will provide
the naming
and contain
we briefly
Computers
77 subroutines
linear
more portable but
of structured
and control
also look at the effect
computational
FORTRAN
and to run efficiently
for linear-equation
on serial machines,
and vector
proposed
to provide
efficiency,
and
a number
on a matrix
We
The library will be based on the well-known
EISPACK
computing
of
of simultaneous
make codes
and increase
performance.
the proposed
which will be freely accessible
code development,
of structure
linear-al-
and Z. Bai.’
and matrix eigenvalue
is intended
of structured
for these problems,
of an algorithm
to be a collection
of various
suffer
dealing with them is being
Library for High-Performance
outlines
fast algorithms
of the sensitivity
for such a matrix.
Ducroz, Ed Anderson,
This minisymposium
parallel
so-called
the
or in other
which then results in complete
in algorithms
available
stability
for
with exploiting
of the problem,
of digital signal processing
constraint
defined
in
derived
answer.
understanding
in which problem
structure
tures,
of these
and of the error propagation
and indicate
solution
concerned
execution,
from the correct
of a correct
have been
more and more these days. Here we first consider
matrices
which
mainly
Yet several
during their (real-time)
and are encountered
algorithms
in order to improve the complexity
words to speed up the algorithm.
should
constraint.
In linear
application
and show
at hand and how algorithms
have been
various
CARLSON
some of these problems
of the problem
be adapted in order to cope with the structure
TAM, AND DAVID
problems,
LINPACK
and
and linear least
infrastructure to exploit
architeccomputer
for scientific
the profusion
of
available. for the routines,
on the structure
give listings
of the routines
for a few
and choice
of
In addition, a discussion of the aspects of software design is given.
‘Department of Computer Science, University of Tennessee, Knoxville, TN 37996, Oak Ridge National Laboratory, Mathematics Science Section, Oak Ridge, TN 37831.
and
717
AUBURN 1990 CONFERENCE ON MATRIX THEORY A Summary
of Research on Linear Algebra and Matrix Theory in Spain
by Vicente Her&&z7 This talk is devoted working
on topics
to giving a picture
related
paid to the research
with matrix
in the Universidad
of the different
theory
PolitCcnica
Some Recent Results on Singular-Value by Roger Horn.’ A quasilinear representation clearer
understanding
hold between inequality
from which
Special
attention
is
de Valencia.
invariant
norms
Ky Fan domination
for all unitarily
many known
groups in Spain which are algebra.
lnequulities
for unitarily
of the classical
two matrices
and linear
invariant
on matrices
theorem
norms.
and new inequalities
leads to a
on inequalities
that
It also leads to a master
can be extracted
as special
cases. Basic notions essential
of duality play a key role in obtaining
in deriving
an apparently
ucts that obey a fundamental
new characterization
majorization
to treat the ordinary and Hadamard generalization
of both products
inequality.
products
cannot
FFTs and the Sparse-Factorization
our results,
and they are also
of those bilinear
matrix prod-
This characterization
permits
us
in a unified way and shows why a natural
satisfy the basic inequality.
Idea,9
by Charles van Loan.” The FFT Algorithms subscript
literature
tend notations.
sen block-matrix Borrowing framework DFT
is vast, disconnected,
to be
detailed
ideas
from
“sparse factorizations.” and its connection
selected FFT
authors,
algorithms.
with
an array of tricks.
obscure
this situation
I have developed The central
of sparse matrices.
multidimensional through
a well-cho-
The theoretical
vehicle
of this activity
FFT algorithms.
has important
a high-level,
unifying
idea is the factorization
Different
FFTs
correspond
that surface
of the
to different
for doing this is the Kronecker
with the kind of data transpositions
fringe benefit
factorizations
level
notation.
for describing
of vector/parallel
and (to an outsider)
scalar
This talk is about how to correct
matrix into a product
important
at the
in FFT
product work. An
is that our notation facilitates the development
So once again we see that the language
computational
overtones.
‘Dpto. Sistemas Informaticos y Computation, Camino de Vera, s/n, 46071 Valencia, Spain.
of matrix
Notation is everything.
Universidad Politecnica
de Valencia,
‘Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218 with Roy Mathias and Yoshihiro Nakamura).
(jointly
‘This paper gives an overview of the book Matrix Frameworks for the Fast Fourier Transform, SIAM, Philadelphia, 1992. “Department
of Computer Science, Cornell University, Ithaca, NY 14853.
718
FRANK
UHLIG,
TIN-YAU
TAM, AND DAVID
CARLSON
Matrix Complements by Carl Meyer.” The
purpose
is to introduce
complementation
tion is focused on stochastic a partitioned
some variants
in block-partitioned
irreducible
matrices
complementation,
stochastic
concept
special
of Schur
structure.
Atten-
which, in its simplest form, is defined on
matrix
Pll
P=
p12
p
21
i with square diagonal blocks.
of the well-known which possess
The stochastic
p22
1
complement
of Pii in P is defined to be the
matrix
sij = Pii Stochastic as important
+ P,j(
complements stochastic
complementation
z-
for
Pjj) - lPji
i = 1,2 and j = I,2.
have a variety of interesting
interpretations.
are discussed,
Several
theoretical
of the algebraic
and then applications
properties
aspects
as well
of stochastic
to Markov-chain
problems
are
developed. Extensions
of these
ered. The concept
ideas to general
nonnegative
of Perron complementation
well as its applications
Lijwner-Ordering
irreducible
matrices
are consid-
is put forth, and some of its properties
as
are presented.
Monotonicity
and
Convexity
Properties
of
Some
Matrix
Functions by K. Norah-iim.‘2 A survey
is given
some matrix functions the statistical
of LGwner-ordering encountered
literature
set of Hermitian
Combinatorial
monotonicity
in statistics.
only for nonnegative
and convexity
Some of these properties,
definite
matrices,
properties
of
considered
in
are here extended
to the
matrices.
Perron-Frobenius
Theory
by Hans Schneider.13 Combinatorial
spectral
a matrix to its spectral
“Department
theory is the study of the relation
properties.
of Mathematics,
of the graph (or pattern)
In this talk, we mainly consider
(reducible)
of
nonnega-
North Carolina State University, Raleigh, North Carolina,
27695. “Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland; current address: Institut fur Mathematik, Universitgt Augsburg, Memminger Str. 6, D-8900 Augsburg,
Germany.
13Department
of Mathematics,
University
of Wisconsin,
Madison,
WI 53706.
AUBURN
1990 CONFERENCE
tive matrices
ON MATRIX
(or, equivalently,
singular
M-matrices)-as
relate the graph of such a matrix to the structure spectral
radius.
We consider
properties,
such
particular
interest
(combinatorial)
positivity
are known.
possible
relations
between
height
Many conditions
We give a solution
by the title. eigenspace
of the eigenspace
of the Jordan
of the (spectral)
level characteristic.
teristics
part
indicated
of the (generalized)
properties
as the corresponding is the relation
719
THEORY
and its spectral
normal
(Weyr)
We for its
form.
A topic
characteristic
of
to the
for the equality of the two charac-
of a long-standing
problem
to characterize
all
the two characteristics.
Combinatorial Aspects of Multilinear Algebra by Jose Dias Da Silva. l4 We report Other cerning
on some recent
combinatorial
aspects
the permanent
results
on the multilinearity
of multilinear
spectrum
algebra
and the spectrum
On the Equality of the Ordinary
partition
are mentioned, of matrices
Least-Squares
of a character.
namely,
associated
those con-
with graphs.
Estimator and the Best Linear
Unbiased Estimator I5 by George Styan.”
It vector
known
is well
in the general
that the ordinary Gauss-Markov
least-squares
matrix V can be the best linear unbiased the identity
matrix.
dispersion izations
of the OLSE
of these conditions,
Block
We also consider
similarity
It comes
single square
“This
Zyskind.
the situation
of the mean
definite
dispersion
even if V is not a multiple perspective,
the model matrix
is a generalization
matrix
X nor the C.
several simple character-
and a rather
complete
set of
when all or part of X is allowed to vary.
of the usual similarity
of square
matrices
a matrix pair (A, B),
A square,
rather
than a
appeared
in the
A. The first studies on this equivalence
talk is based
just published
We present
of
the development
Problems on Block Similarity
up when we consider
“Department Canada
Neither
along with various examples
“Department of Mathematics, 30 Piso, 1700 Lisboa, Portugal. Finland)
in a historical
to be BLU.
Rao, and (the late) George
Some Outstanding by Ion Zaballa.‘? fields.
(OLSE)
matrix V need be of full rank. The key results are due to T. W. Anderson,
Radhakrishna references.
(BLU) estimator
In this talk we discuss,
of the several conditions
estimator
linear model with nonnegative
Universidade de Lisboa, Rua E. Vasconcelos,
on a joint
paper
(with discussion
of Mathematics
relation
and
with
Simo
and rejoinder) Statistics,
Puntanen
in Amer. McGill
(University Statist.
University,
over
Bloco CL
of Tampere,
43:153-164
(1989).
Montreal,
Quebec,
H3A 2K6.
17Department Vitoria-Gasteiz,
of Mathematics, Spain.
Facultad
de Farmacia,
Universidad
de1 Pais Vasco,
01007
720
FRANK UHLIG, TIN-YAU TAM, AND DAVID CARLSON
control
theory literature
and dealt with linear time invariant
systems of the form
f+) =Ax(t) +h(t). In this context
the equivalence
we say that block-similarity pencils
is usually called feedback
or feedback
equivalence.
In matrix theory
is the strict equivalence
of singular
of the form [sZ, - A, - B].
During relation.
the last years a number
This talk presents
OF
CONTRIBUTED
Extreme
Points of a Set of Positive
by William Fairleigh George
N. Anderson,
Dickinson
have been working
on this equivalence
results as well as some open problems.
TALKS”
Semidefinite
Matrices
Jr., Department
University,
Teaneck,
of Mathematics NJ 07666
and Computer
Science,
(jointly with T. D. Morley and
E. Trapp)
Operators
by LeRoy
Preserving
B. Beasley,
UT 84322-3900 Inversion
of people
some of the obtained
3. LIST
Linear
equivalence
Idempotent Department
Matrices
over Fields
of Mathematics,
Utah State University,
Logan,
(jointly with N. J. Pullman)
of Infinite
Matrices
by Kerry G. Brock,
Department
of Mathematics,
Georgia
Institute
of Technology,
Atlanta, GA 30332-0160 On the Convergence by Jack 36849 The
B.
of the Iterative
Brown,
Proportional
Department
Fitting
Procedure
of Mathematics-FAT,
Auburn
University,
AL
fjointly with P. Chase and A. Pittinger)
pth Roots of a Matrix by Bryan 50011
Elementary
Cain,
Department
Divisors
and Ranked
by Keith L. Chavey, son, WI 53706 Circularity
Iowa
State
University,
Ames,
IA
of Mathematics,
to Matrix Compounds
University
of Wisconsin,
Madi-
(jointly with R. A. Brualdi) Chien,
111, Republic
Study of Complex
Posets with Applications
Department
of the Numerical
by Mao-Ting Taiwan
of Mathematics,
(jointly with T. Laffey)
Range Department
Symmetric
by Dipa Choudhury, 4501 N. Charles
of Mathematics,
Soochow
University,
Taipei,
of China Matrices
Department
Str., Baltimore,
of Mathematics,
Loyola
College
in Maryland,
MD 21210-2699
‘*Only those are listed here that are not represented as papers in this issue.
AUBURN
1990
CONFERENCE
Determinantal
Inequalities
ON MATRIX
for Positive
Cox rg, Department
by B. Ann
721
THEORY
Definite
Matrices
of Mathematics,
Brigham
Young
University,
Provo,
UT 84602 Diagonahzing
the Adaptive
by Jerome Park, Some
Dancis,
Interlacing Apartado
Results Leal 3008,
Idempotence, Atlanta, Modified
3000
Hermitian
Department
Coimbra,
University
Acyclic
Iterative
Methods
(jointly
and Sign-Pattern
New
Lyngby,
State
University,
of Spectral
McEachin,
Ordering
Systems Ohio
and Pseudo
Institute,
at Fort
Wayne,
of Generalized
of Mathematical Fort
Wayne,
Inverses
of Almost
malai
Tamil
India
6008002,
School
GA 30332-0160
Nadu,
the Resolution
University,
of Mathematics,
(jointly
n’Ekwunife
Systems of Denmark,
Sciences,
Indiana
University,
Definite
Matrices
Annamalai
University,
Anna-
P. 0.
with
Georgia
Institute
of Technology,
Atlanta,
W. L. Green)
of E. Dittert’s
Conjecture
Muoneke,
Department
Drawer
125, Prairie
of Mathematics, View,
Prairie
View
A and
M
TX 77446-0125
Pivot
Larry
Neal, City,
Permutation New
Department
of Mathematics,
TN 37614-0002
(jointly
with
East
Tennessee
State
University,
G. Poole)
Matrices
by Christopher Bedford
University
of Operators
by T. D. Morley,
Johnson
45701-
IN 46805-1499
of Mathematics,
The Rook’s
OH
Variation
Department
Department
Intervals
Athens,
Gyroconservative
The Technical
by A. R. Meenakshi, Nagar
University,
and L. Snyder)
Symmetrizable,
Mathematical
University
On Partial
Similar
Georgia
Denmark
Examples
Purdue
Linear
of Mathematics,
A. Gunawardena
Kliem,
by Raymond
by
de Coimbra,
Matrices
of Mathematics,
for Consistent
of Normalizable,
DK-2800
by
Universidade
Portugal
Department
Department with
by Wolfhard
Towards
College
GA 30303
The Dynamics
Order
of Maryland,
Matrices
of Mathematics,
Idempotence,
A. Eschenbach,
by S. K. Jain, 2979
for Indefinite
Duarte,
Qualitative
by Carolyn
Some
Method
of Mathematics,
MD 20742
by Antonio
Some
SOR Iterative
Department
W. College,
Norman,
Department
Egham,
Matrix
Results:
Generated
lgCurrent 36849-5307.
address:
Department
Surrey
TW20
of Mathematics, OEX,
by Two Statistical
of Mathematics-ACA,
Royal
Holloway
and
England
Applications
Auburn
University,
Auburn,
AL
722
FRANK UHLIG, by Patrick 76798-7328
L. Odell,
On the Commutant Vector Space by Christian 2800
Department
of a Linear
Mechanical
Baylor University, Operator
Department,
Danish
Waco,
on a Finite
TX
Complex
Engineering
Academy,
College
of William
and Mary,
College
of William
and Mary,
Lyngby, Denmark
A Review of Recent by Leiba
Results on Inertia-Preserving
Rodman,
Williamsburg, Inertia Theorems by Leiba
Department
for Matrix Polynomials
Rodman,
Approximations
Linear Maps
of Mathematics,
VA 23185
Williamsburg,
Department
of Mathematics,
VA 23185
of the Spectral Radius, Corresponding
Modulus of an Eigenvalue by Uriel Rothblum, University,
RUTCOR,
between
Ill-Conditioned by Burkhard Columbus,
Eigenvector,
for a Square Nonnegative
New Brunswick,
On a Relation
Robust
Rutgers
Schalfrin,
Irreducible
Center
and Second Largest
Matrix
for Operations
Research,
Rutgers
NJ 08903 Prediction
Linear Equation
and Accelerated
Iteration
Schemes
for
Systems
Department
of Geodetic
Science,
Ohio State University,
College
of William
OH 43210-1247
On Inversion of Symmetric by Tamir
Shalom,
Williamsburg, On Perturbations
Toeplitz Matrices
Department
of Mathematics,
VA 23187-8795
and Mary,
(jointly with L. Rodman)
and Transformations
by Nagwa Sherif, Department 13060
of Mathematics,
and Anticommulant Pommer,
TIN-YAU TAM, AND DAVID CARLSON
of Orthogonal
of Mathematics,
n-frames Kuwait University,
P.O.
Box 5969,
Safat, Kuwait
Parallel Sparse LU Decomposition by A. F.
van der
Amsterdam,
P.O.
Staapen,
Box 3003,
on a Mesh of Transputers Department 1003
MSE,
Koninkijke/Shell
AA Amsterdam,
The Netherlands
Laboratorium Cjointly with
R. Bisseling and J. G. van de Vorst) Matrices That Commute
with a Generalized
by Jeffrey L. Stuart, Department sippi, Hattiesburg, An Extension
Permutation
of Mathematics,
Matrix University
of Southern
Missis-
MS 39406-5045
to Outer Products
with a Variety of Applications
by Tien-Hsi Teng, 12D, Maple Mansion, Taikoo Shing, Hong Kong On a New Eigensolver by Anna Tsao, MD 20715-4300
for Real Diagonalizable
Supercomputing
Research
Matrices with Real Eigenvalues Center,
On the Structure of the Product of Boolean E-Functions by Gene Underwood, Department of Mathematics, UT 84322-3900
17100
Science
Drive,
Bowie,
Utah State University,
Logan,
(jointly with Steven Lederman)
AUBURN
1990 CONFERENCE
More on the Reverse-Order
ON MATRIX
Law
by Hans J. Werner,
Institut
Bonn, Adenauerallee
27-42,
Symmetric
Hankel
Operators:
4.
La Jolla, CA 92093,
SYNOPSES A DIXIE CUP:
consider
We
the
set of all
symmetric
for future
n
Universitlt
(jointly with J. W. Helton)
INTERVALS
x
real
n
convenience.)
OF MATRICES
and G. E. TRAPP23
symmetric
space with inner product
as an
n(n + 1)/2-
(A, B) = tr(AB)/2.
matrices
(The factor
We shall use the following
basis of the real
2 X 2 matrices:
A,=
Note that this is an orthonormal
[
;
_;
1 ,
basis with respect
If A and B are n x n real symmetric
matrix is termed
denote
n
the
set
of
positioe
n
x
symmetric
B - AEPos,.
The set Pos is a cone, i.e.,
X > 0. Thus
A 6 B is the sort of order
and
1 1
A,=
;
:,
to the above inner product.
matrices,
for all vectors x. A real symmetric
relativity:
Research,
Completions and Eigenstructure of Mathematics, University of California,
JR.,21 T. D. MORLEY,22
real inner product
of two is chosen
und Operations
Bonn 1, Germany
VISUALIZINGORDER
by W. N. ANDERSON,
dimensional
fur 6konomie D 5300
Minimal-Norm Department
by Hugo J. Woerdemans’ San Diego,
723
THEORY
we say that A < B if Ax
* x < Bx * x
positioe if A 2 0. We let Pos = Pos,
matrices;
then
A Q B if and only if
Pos + Pos C Pos, and hPos C Pos for all real that one usually
encounters
first in special
if we think of A + Pos as the light cone of future events to A, then A < B if
and only if B is in this cone. To visualize
the cone Pas,,
we simply note that
X = ArA, + AsA, + AsA, =
h
+
x,
h
A3
3
is positive
A,
-
h
1
if and only if Xi + hs > 0, X, - hz > 0, and A; - % - A” 2 0. Putting these
20Current address: Department burg, VA 23185.
of Mathematics,
21Department of Mathematics Teaneck, NJ 07666.
and Computer
22School of Mathematics, morley~cerc.uvu.uMet.edu. 23Department town, WV 26506;
of Computer
Georgia Science
[email protected].
Institute
College of William and Mary, WilliamsScience, of
and Statistics,
Fairleigh
Technology, West
Virginia
Dickinson Atlanta,
University, GA
University,
30332; Morgan-
724
FRANK
conditions
together,
UHLIG,
we see that the cone
TIN-YAU Pas,
TAM, AND DAVID
CARLSON
is simply the geometric
((A,, &> A,) I Al 2 0, AI 2 22 + A?,. If A is a positive symmetric matrix, we define the order inter&,
cone
denoted
{X =
[O,A], by
[O,A] = {X]O
study
generalized (with respect
where
to a suitable
A,, : Y-
e.g., A,, Either
the structure
of [O,A], we need
Y,
the matrix
matrix viewpoint
Ai2:
basis) write A as
YL++
etc.
Y,
viewpoint requires,
of shorted
(If we think of A as a linear where P is the orthogonal
or the linear-operator
however,
bases.)
where dagger denotes
to Y,
operator
operator
the Moore-Penrose the (1,l)
dimensions,
alternative
are needed.
definitions
then we define the parallel We refer the reader
can be expressed
pseudoinverse.
the same as A: then
We also need the concept
viewpoint
entry
then
onto
can be adopted.
Y. The
track of the
as
If A is invertible,
of Y(A)
or
we can
operator,
projection
that some care be taken in keeping
The shorted
partitioned
and A-’
is ([A-‘],,)-l.
is
In infinite
of paraZle2 sum. If A and B are two positive
matrices,
sum A : B as A : B = A(A + B)+B.
to [l, 21 for the basic facts about the shorted
operator
and the
sum.
If VE Pos is a convex set, i.e., an
concept
orthogonal
is simply PA restricted
various orthogonal
parallel
the
Schur complement. If A is a positive matrix, and Y is a subspace,
XV+
extreme point of V if whenever
then Y = Z = X. Intuitively,
(1 - A) VS
@? whenever
0 < X Q 1, then X is
X = hy + (1 - A)Z E $? for some Y and Z in Y,
the extreme
points of V are the corners
theorem
THEOREM
Let A be a positioe n x n real matrix. Then X E [O,A] is an extreme conditions are satisfwd:
1.
is from [3]; see also [7, 91 for related
of V.
The following
results.
point of [O,A] if and only if any of the foU&ing
(a) Then matrix X is the shorted operator of A to some subspace Y. (b) X : (A - X) = 0. (c) X = A1f2PA1/’ for some orthogonal projection P. (d) range(X)
tl range(A - X) = (0).
The above theorem
allows us to give a picture
Let +A denote the linear function
of [O,A]. Suppose that A is invertible.
@A : X - A1/‘XA1/‘. Note that a;’
= +*-I.
It is easy
to see that @A maps [O,I] to [O,A]. Thus the convex structure of [O,I] is the same as the convex
structure
of [O,A]. But the shorts of I are precisely
the orthogonal
projections
AUBURN
1990 CONFERENCE
FIG. 3.
(see [l]); indeed,
ON MATRIX
A polygonal approximation
to the set [O,I] for 2 X 2 matrices.
the fact that the orthogonal
projections
is known independently
of shorted
operators
projections
Ps =
Expanding
I 1 cos 8
sin 8
[COST
p = tI
extreme
We now consider
matrices)
we notice that the rank 1
co?e
sin e cos 8
sin e cos 8
sinse
1
sin@] =
and using trigonometric
A, + (sin20)Aa
we get
+ (cos20)Aa
points of [OJ] are a geometric
the intersection
of the two equal expressions
of two order intervals.
circle. The following result is due
(see [2]), we write A : B : C for either
(A : B) : C or A : (B : C).
24The figures in this paper from the second
identities,
I’
2
to T. Ando [4]. Since the parallel sum is associative
available
points of [O,I]
are of the form
out the matrix elements
Thus the rank-one
are the extreme
(see [8]).
To see this picture 24 (for the 2 x 2 symmetric orthogonal
725
THEORY
author
were produced via E-mail.
by MathematicaTM.
The computer
codes
are
726
FRANK UHLIG, TIN-YAU TAM, AND DAVID CARLSON THEOREM 2.
Let A and B be two positive n x n real matrices, and set V = ‘8”. B
= [O,A] n [O,B]. Then the matrix X E V is an extreme point of V ifand following
conditions
(a) range(A - X) fl range(B (b) (A - X):(B
- X):X
The above theorem
- X) Cl range(X)
either
allows us to explicitly
of V=
therefore
construct
semidefinite
the extreme
matrices.
points in the case
If either A or B is singular,
%“,B = [O,A] fl [O,B] re d uces to the one-dimensional
A < B or B < A, then
[O,A] n [O,B] reduces
that A and B are invertible,
the extreme
= (0).
= 0.
n = 2. Let A and B be 2 x 2 positive then the geometry
only ifany ofthe
are true:
and that neither
to the previous
case.
case. If Assuming
A < B nor B < A, we can classify
points of V by their rank:
Rank 0:
The zero matrix.
Rank 1:
For each angle 0, let Y0 Since ble.
be subspace
spanned by the vector
[cos 0, sin OIT.
YO(A) and Y@(B) are rank one with the same range, they are comparaThe
operators
min{ Y@(A), Y@(B)}
are precisely
the rank-one
extreme
points. Rank 2:
The set of 8’s for which min{ Y@(A), YO(B)} = YO(A) is an interval, For each such 0, let &, be chosen B}.
Then the rank-two
extreme
points are precisely
as 0 ranges over 0 E (@o, 0,).
At the extreme
extreme
in this
points
constructed
[0,, 8,].25
to solve max{ A( YO(A) + h[ A values of 0, i.e.,
manner
Y@(A)] <
Y@(A) + b[ A -
reduce
YO(A)]
B0 and 8,, the
to rank-one
extreme
points. The above theorem construction extreme
generalizes
points generalizes
modifications.
to a construction
The
of the rank-n
points.
We now work out a numerical write (x,
to the n x n case, with suitable
of the rank 2 extreme
y, z) for the matrix
example.
xA,
Let A,, A,, and A, be as before, and let us
+ yA, + zAa. Let A and B be the matrices
A = (2,1,0)
= 2A, + A, =
and B = (2, -
1,0)
= 2A, - A,=
Then [O,A] n [O,B] is, in terms of coordinates, (z - 2)2 > (y -
I
:,
the intersection
;
I
. of the cones
1)2 + z2,
(x-2)2~(y+1)2+z~, x2 2 y2 + 22, x > 0.
25Via the correspondence 0 ++eie, we consider angles as points on the unit circle; thus an interval means an interval of the unit circle.
AUBURN
1990CONFERENCE
FIG. 4.
ON MATRIX
[O,A] rl
This is easily solved. The rank-one
and the same with the y-coordinate points is the piece
“triple
the points
727
[O,B], top transparent.
extreme
points are pieces
negated.
The “upper
of the two ellipses
handle”
of rank-two extreme
of the hyperbola y = 0,
between
THEORY
z = $,
(r
- 2)” - 22 = I
y = 0, z = + f.
The ellipses
and hyperbola
meet
at the
points”
We close with a picture
of
[O,A] fl[O,B].
REFERENCES
1 W. N. Anderson, Jr., Shorted operators, SIAM J. Appl. Math. 20:520-525 (1971). 2 W. N. Anderson and G. E. Trapp, Shorted operators II, SIAM J. Appl. Math. 28:60-71
(1975).
728 3
FRANK N. Anderson
W.
and G.
semidefinite operators,
E.
UHLIC, Trapp,
TIN-YAU The
TAM, AND DAVID
extreme
Linear Algebra A&.
points
106:209-217
4
T. Ando, unpublished communication.
5
G. P. Barker, The lattice of faces of a finite-dimensional 7:71-82
6
S.
L.
Eriksson
and H.
Math.
7
W. L. Green
8
Gert
9
of a set of positive
(1988). cone, Linear Algebra Appl.
(1973).
addition,
Sot.
CARLSON
Monogr.
Extreme
C*-Algebras
theoretic
of Operators
approach
points of order intervals,
and Their Automorphism
14, ISBN 0076-0560,
Shorts
A potential
to parallel
(1986).
and T. D. Morley,
K. Pedersen,
Pekerev,
Leutwiler,
Ann. 274:301-317
Academic,
and
to appear. London
Groups,
Math.
New York, 1979.
Some Extremal
Odessa,
Problems,
1989,
to
appear. 10
A. Shapiro,
Extremal
Appl. 67:7-18 11
S. Wolfram, Redwood
INDUCED
problems
on the set of nonnegative
matrices,
Linear Algebra
(1985). MathematicaTM-A
City, Cahf.,
System for Doing Mathematics,
Addison-Wesley,
1988.
NORM OF THE SCHUR MULTIPLIER OPERATOR
by T. ANDOz6 and K. 0KUBOz7 Let
M,, be the vector
[bij] E M,, denote product
An B =
space of all n square complex
by A0 B their
Schur
matrices.
(or Hadumard)
product,
For A = [aij], B = that is, the entrywise
[ aijbij]. Then each A EM, gives rise to a linear operator
called the Schur multiplier norm of S, with respect
operator,
to a norm 1) -
defined
by S,(X)
S, on M,,
= A0 X (X E M,). The induced
)(on M, is defined by 11S,\( := sup 11 x 11 Q111A0 X (1.
A familiar and useful norm on M, is the spectral norm:
another
where
useful norm is the numerical
radius:
w(A)
= s;p
)I .
‘(,;‘;(,;”
\I and ( * ) . ) denote the Euclidean
,
norm and the standard
inner product,
26Division of Applied Mathematics, Research Institute of Applied Electricity, Hokkaido University, Sapporo 060, Japan. 27Mathematics Laboratory, Sapporo College, Hokkaido University of Education, Sapporo 002, Japan.
AUBURN 1990 CONFERENCE
729
ON MATRIX THEORY
respectively. We denote the induced norms of SA with respect to the spectral norm and the numerical radius by 1)S,ll_ and 11S,ll w, respectively. It is mentioned in [2] that Haagerup succeeded in determining IIS, II_ in the following form. For A = [aij] EM,,
HAACERUP’S THEOREM.
the following
assertbns
are mutually
equivalent : A = B*C such that
(ii) A admits a factorization
B*BoI&Z
C*CoZ
and
where Z is the identity (or unit) matrix. (iii) There are vectors 2, iji E G” (i = 1,2, . , . , n) such that 1,2, . . . , n) and aij
=
(iv) There are 0 Q R,, R, E M,,
(Zjliji)
(i,j
= 1,&n).
such that
R,oZ
and
In this note we are going to give the characterizations derive Haagerop’s theorem for it as a consequence. THEOREM.
(9, (ii),
II?ziI(, I( GilI Q 1 (i =
R,oZ,
of the norm I(S,II w, and to
For A = [ aij] E M,, the following assertions are mutually equivalent:
II%ll, G 1. A admits a fwtorieation
A =’ B*WB such that B*B-Z&Z
(iii)w There
are vectors
II$1) Q 1 (i = 1,2,.
?ieO”
(i = 1,2,.
. . , n) and a matrix
. . , n), W*W Q I, and aU = (WGjlZi)
(iv), There is 0 < REM,
W*WQZ.
and
(i = 1,2,.
..,n).
such that
[
: 1 A R
20
and
RoZgZ.
WE M,, such
that
730
FRANK
UHLIC,
We can derive from the theorem Schur multiplier (I)
llS,ll,
TIN-YAU
the following
TAM, AND DAVID
properties
CARLSON
of the induced
norms of
operators: C IIS,ll,
Q 2llS,ll0,
(AeM,).
(2) llS,ll, Q II AlI, (AEM,). (3) I]S,]] oD= I(S,]] w if A is Hermitian. (4)
]IS,I] m = ]]S,]] w = 1 if A is unitary.
(5)
llS,ll, G llSI~~+~~*~ llw (AEWJ
(6)
](S,]],.(]SIA~)]u;if
Aisnormal.
The details will he published
in [l].
REFERENCES 1
T. Ando and K. Okubo, Induced
2
Appl., to appear. V. I. Paulsen, Completely Bounded Maps and Dilation, Math.
146, Longman,
Essex,
norms of Schur multiplier
U.K.,
operator, Pitman
Linear Algebra
Research
Notes in
1986.
AN APPLICATION OF VALUATION THEORY TO THE CONSTRUCTION OF RECTANGULAR MATRIX FUNCTIONS by
1.
JOSEPH A. BALL and MAREK RAKOWSK12’ Zntroduction The problem
functions
we address
with a given
complex
plane C,2
B
formed
m
X n
by functions
rational
m x n rational
matrix
and
function
WmXn(u)
WE gnx”,
analytic
functions.
do there
with m
x
exist rational
subset
matrix
c of the extended
exist, how to find one? and by B(u)
the subring
will denote the g-vector on u. Bnx” B? mXn (u) will denote the %?(o)-module
which are analytic
algebraically find
When
over a proper
the field of scalar rational functions
matrix functions
can be characterized Bmx”
structure
If such functions
We will denote by 9 of
here is as follows.
zero-pole
Nw, fiwe
over
BmX”(u),
of of
on 0. The zero and pole structure
in terms of coprime
n matrices
space space
B
factorizations. and
g(u),
Namely, identify
respectively.
DWs kJfflx”(u), and fiw~
Given a gmxm(u)
such that (i) det Dw # 0 and ew,
NW are right coprime
(ii) det D, # 0 and D,, NW are left coprime (iii) W = N,D&’ = fi&‘6w. In this notation, functions
[over g(u)], [over g(u)],
W, HE 92 lnxn have the same left zero structure if NH = NwQ W and H have the same right pole structure
for some Q such that Q, Q- ’ E %! “x”(u).
28Department of Mathematics, Virginia Tech, Blacksburg, VA 24061.
AUBURN
1990 CONFERENCE
if fin = Rfiw
for some
THEORY
731
R such that R, R- ’ E .%’“.“(u)
(see [2] for the regular case,
that is, when the functions vanish
ON MATRIX
involved
are square
and
determinants
which
do not
identically).
A concept null-pole
which
refines
subspace
the notion
W over
of
WS? nx1( a). For the regular
u,
case,
l] (see [2] for a comprehensive matrix functions have been Clearly,
functions
of zero
i.e.
null-pole
in S? mx”
subspaces
the same
subspaces For matrix
over some
functions
methods
right
pole
have played
u is that
9?‘nx1,
of a
yO(W) =
a key role in [7, 8, 3, rational
over u have the
over u. Indeed, if W, HE 9 mx”, then Q such that
left zero
Y,(W)
=
(see Proposition
Q, Q- ’ E W mxn(u)
and HE W mx”ff, then W = HQ for some
structure
S’rm pl e examples over
u may
have
show
that
different
functions null-pole
u. time
the
analytic
has been
for finding
been
and
over of
Null-pole subspaces of nonregular
Q E S? “*x”“( u) if and only if J$( W) C Y,(H). with
structure
with the same null-pole subspace
if WE .%’n’x”u
generally,
and pole
W(u)-submodule
in [6].
investigated
Yo( H) if and only if W = HQ for some 1.1 in [4]). More
the
exposition).
same right pole and left zero structure
have
have
a regular
available.
description
known
One
rational
subspaces
matrix
function
with
there). a given
of regular
rational
Also, constructive null-pole
subspace
to the general case is via embedding in the regular
approach
case. A tool for such embedding
of null-pole
(see [2] and the references
is provided by valuation theory (see [9], [12], and, for
the general theory [ll]).
2.
Orthogonality Let
from
in 2 n
h be a point
of the
where
r) is the unique
integer
r(z)
with
F analytic
complex
stronger
triangle
such
plane.
We define a function
( * 1z=h
=
that
1
(Z-X)“?(z)
if
XeC,
z-V?(z)
if
X=03
at h. The function
and nonzero
1n ) z=h < 1 for every
integer
n, the valuation
) . ) ==Ais a real valuation of W . Since ) * 1z=x is non-Archimedean and the
inequality
I r1 holds
extended
S? into the set of real numbers by putting
for all rr, rs E 9.
+ r2
I z=x Q m=4 I t-1 I z=h, I f-2 I z=hl
732
FRANK UHLIG,
TIN-YAU TAM, AND DAVID CARLSON
Let n be a positive integer and let he C,. the product
We define a function
)] .
1)z=x on W”,
of n copies of ~8, by putting
ll(rIpf-2,.
..,
rn)Ilz=x= m={lrll.=~, lr21z=h....,ImldJ
In this way 9? ” becomes a normed vector space over the real valued field ( 1, Following
the definition of orthogonality
shall say that two subspaces
in a non-Archimedean
A and Q of W * are orthogonal
I * ( ==,,_).
normed
at XE C,
space, we
if
IIx + Y II==h= m={ IIr IIdI II Y IIz=h} for each
r E A, y E Q. We shall say that A and Q are orthogonal
orthogonal
at every point of u. We shall say that vectors
at
(respectively,
XE C,
on a subset
{xi:i#j)areorthogonalat
X(on
o of C,)
u)forj=
W” and AEC,,
Ifhisasubspaceof
on u C C,
if they are
rr, ~2,. . . , rk are orthogonal
if the spans over
W of { xj}
and
1,2,...,k. we will denote by A(A) the set of values at X
of those functions in A which are analytic at A. Plainly, A(A) is a subspace of C”. The space A(A) can be characterized coefficients
equivalently
as the linear span over C of the leading
in the Laurent expansions at A of the functions in A. Using this notation, we
can characterize
orthogonality
in 8 ” as follows (see Proposition
2.3 in [6]).
Let A and Q be two subspaces of W “, and let A E C,.
PROPOSITION2.1.
Then A
and R are orthogonal at A if and only if A(A) fl R(A) = (0). It follows from the definition of orthogonality at a single point AEC, subspaces
of
9”.
necessarily
that two subspaces of W ” orthogonal
have the trivial intersection.
We say that the subspace
Q is an orthogonal
subspace A in (C, a) if A and n are orthogonal the next proposition,
see Proposition
PROPOSITION2.2. proper
subset
u of C,.
Let A, 0, and C be of the
complement
on o and A + Q = X. For the proof of
2.5 in [5].
Let A and Q be subspaces Then Q has an extenskm
of 92 n which are orthogonal
on a
complement
of A in
c C C. Also, Wax”
will
to an orthogonal
(W”, 0). 3.
Analytic Description of Pole and Zero Structure For notational convenience, we will assume hereafter
denote m x n rational matrix functions analytic in C-1
u and vanishing at infinity. By
partial fraction expansions, we have an exact sequence
where all vector spaces are over C. We will use the symbol I’,= for the projection (3.1) for arbitrary positive integers
m and n.
as in
AUBURN
1990 CONFERENCE
If WEWmxn, C-linear
space
ON MATRIX
we define
P,c( yO( W)).
the
733
THEORY
right pole structure
of W over
8,
For th e p roof of the next proposition,
o to be the
see Proposition
4.1
in [6]. PROPOSITION3.1.
8,
= {C,(z The
Plainly,
pair (C,,
A,) in Proposition
denote
the left annihilator
is any matrix polynomial Let
3.1
is called
a right pole pair
such that
for W over
(I.
C u.
u( A,)
Let W”
There exists an obseruable pair of matrices (C,, A,) : x a constant vector}.
- A,)-‘x
of W in g1 xm. A
Zdt kernel polynomial of W
whose rows form a minimal polynomial
XE u be a zero
of W. Choose
(a lxm,X). One can show (see Proposition of matrices (A, Bx) such that
P,c({~En,:4WcWlx”(c)})
an orthogonal
basis (see [9]) for W”.
complement
Ax of W”l
4.3 in [S]) that there exists a controllable
= (~(.a
- A,)-‘Bx:
in pair
z isaconstantvector}.
If x,, &, . . . , X, are the zeros of W in u, the pair
diag(Ax,,A&
,...,
Aht),[B$
BE
a..
B<’ 1)
is called a lej? null pair for W over u. PROPOSITION3.2. One canjnd such thatPO~({q5~A:~W~W1Xn(u)}) For the proof of Proposition Proposition
an orthogonal complement A of W” in ( glx”‘, = {x(z -A,)-‘B,: raconstantuector}. 3.2,
see Lemma
3.16
3.2 a subspace associated with the pair (At,
If (C,,
A,),
(Al.
a)
in [5]. We will call A as in
Br).
Br), and P, are as above, there exists (see Theorems I’ such that rA, - ATr = BrC, and
2.7, 3.1, and
3.3 in [S]) a unique matrix
%(W)
=kerP,n
C,(z-A,)-‘x+h(z):+~C”‘~~,h~W~~~(u),
and c
Res,,,O
(z - A,)-‘B,h(
z) = TX
Z&l A triple ((C,, A,), (AC, Br), r) is called a I& u-spectral triple of W.
4.
Construction A construction
triple
of a Function
is given in [lo].
function Theorem
with prescribed 6.2 in [6].
with the Prescribed
of a regular rational matrix function Below,
we utilize
left u-spectral
Null-Pole
Subspace
with a prescribed
it to construct
a rectangular
triple and kernel polynomial.
left u-spectral rational
matrix
For the proof, see
734
FRANK THEOREM 4.1.
given. left
Let
matrices
Then there exists a rational a-spectral
conditions
0)
triple
and
P,
UHLIG, C,,
A,,
TIN-YAU Ay, B,, r
matrirfunction
TAM, AND DAVID and
with
polynomial
as a left kernel
a matrix
7, = ((C,,
polynomial
A,),
g and
CARLSON P,
be
( Ay, Bc), I’) as a
only
if the following
hold:
the pair
(C,,
A,)
(ii) t& pair (At.
is obseruable
and u( A,)
C u,
and u( Ar) C u,
Br) is controllable
at infinity,
(iii) P, has no zeros in C, and its rows are orthogonal
(iv) the rational matrix function PK( z)C,( z - A,)- ’ is analytic on C, h,} and lk is the identity matrix with the same number {A,,&,..., (v) $a(A&= rows as P,, the pair diag( A(, X,1,,
. . . , A&),
[ B;
P(A1)’
...
PN’]
of
T,
is controllable, (vi) I’A,
- A$
= B< C,.
If rS and P, satisfy conditions 7, as a left u-spectral
triple
(i)-(vi)
in Theorem
and P, as a left kernel
4.1, a function polynomial
WE 9?“x”
with
can be constructed
as
follows.
step 1.
Find a regular Choose
Step 2.
rational
matrix function
a Smith-McMillan
Let Y be the largest largest geometric of the geometric
factorization
geometric
multiplicity multiplicity
H with 7, as a left u-spectral
multiplicity
of a pole of H in u, let /J be the
of a zero of H in u, and let q be the largest sum of a pole and the geometric
of H at any single point of u. Let di denote
multiplicity
9ij
manic polyno-
has a zero at a point XE u of order k then
zero at h of order k. Define
=
an m x r) matrix polynomial
1
if
‘i=jQv-p,
Pi
if
v-r
if
i=j-q+m>m-CL,
1
i 0
of a zero
the ith diagonal entry of D. For
v, let pi be the minimal-degree
i=v--+l,q-p+2,..., mial such that if d,_s+i
triple.
EDF of H, and put W, = ED.
pidi has a
Q = [qij] by
otherwise,
and put W, = W,Q. Step 3.
Find a subspace
E associated
with the pair (A,,
W, onto Ker P along the subspace Step 4.
Find an orthogonal U u( AI)),
complement
and a basis
{ul, 02,.
of W m xl
Bt). Project
annihilated
every column of
by E to get W,.
A of the column span of W, in (Ker P, u( A,) . . , ul} for A such that the function [ul up
. . * ul] has neither zeros nor poles on u( A,) U u( A(). Put W, = [Ws 01 ~2 . . . 011. Step 5.
Multiply
W, on the right by a regular
poles or zeros on u( A,)
poles nor zeros in u \ (u( A,) Particular
rational
matrix
U u( Ar), so that the resulting
steps of this construction
function
function
Q, without
W has neither
U a( A()). are illustrated
in [5] with specific
examples.
AUBURN
1990 CONFERENCE
ON MATRIX
735
THEORY
REFERENCES 1
J. A. Ball,
N. Cohen,
improper
rational
and A. C. M. Ran, Inverse
matrix functions,
Matrix Functions (I. Gohberg, J. A. Ball, I. Gohberg,
3
J. A. Ball, I. Gohberg,
and L. Rodman,
problems
matrix functions,
Sot., 4
and L. Rodman,
Boston,
for rational
zero-pole
Two-sided
OT (I. Gohberg,
functions,
submitted
McMillan structure
Birkhbser,
Null-pole
subspaces
G. D. Forney, I. Gohberg
Jr.,
matrix functions
to appear.
of nonregular
rational
matrix
rational
matrix
problems
for rational
matrix
problems
for rational
matrix
of nonregular
spectral
inverse
spectral
Zntegral Equations Operator Theuy 10:349-415
multivariable functions
rational
(1987).
Linear Algebra Appl. 86:237-282
functions,
10
Math.
for publication.
J. A. Ball and A. C. M. Ran, Local
9
interpolation
to appear.
J. A. Ball and A. C. M. Ran, Global inverse functions,
degree
Linear Algebra Appl.,
Zero-pole
Ed.),
J. A. Ball and M. Rakowski,
8
Lagrange-Sylvester
in Proc. Symp. Pure Math., Amer.
Minimal
structure,
J. A. Ball and M. Rakowski, functions,
7
1988, pp. 123-175.
Znterpolatixm of Rational Matrix Functions,
to appear.
with prescribed
6
for regular
1990.
J. A. Ball and M. Rakowski,
5
problems
Ed.), OT 33, BirkhHuser, Boston,
2
OT 45, Birkhguser,
spectral
in Topics in Znterpolatiun Theory of Rational
Minimal
linear systems,
bases of rational
spaces,
SZAMJ. Control 13(3):493-520
and M. A. Kaashoek,
and minimal
vector
(1987).
divisibility,
An inverse
spectral
with applications
to
(May 1975).
problem
for rational
matrix
Zntegral Equations Operator Theory 10:437-465
(1987). 11
4. F. Monna,
12
G. Verghese
Analyse Non-archimdienne, and T. Kailath,
Rational
Springer-Verlag,
Matrix Structure,
New York, 1970.
in Proceedings of the 18th
ZEEE Conference on Decision and Control, Fort Lauderdale, FL,
Vol.
2, Wiley, New
York, 1970, pp. 1008-1012.
CLASSES OF STABLE MATRICES by ABRAHAM
BERMAN2’a
The inertia, i+(A)
3o and DAFNA
in A, of a square
is the number
of pure imaginary
matrix
of eigenvalues
eigenvalues
SHASHA2’
A is a triple
(i+(A),
io( A), i_(A)),
where
of A in the right open half plane, io( A) the number
of A, and i_(A)
the number
of eigenvalues
in the left
open half plane. A matrix
A E R”* ”
is stable for every positive
diagonal matrix (a diagonal matrix whose diagonal entries
“Department
of Mathematics,
IS (positive) stable if i+(A) = n. A is D-stable if AD
Technion-Israel
Institute
of Technology,
Israel. 30Research
Supported
by the M. and M. Bank Research
Fund.
are
Haifa 32000,
736
FRANK
positive) matrix
D.
A is (Lyapwwv)
D such that
diagonally
diagonally
are D-stable.
in differential
5, 4, 8, 91. A real matrix
study these
preserving
are D-stable.
[semildefinite.
Stable,
equations,
TAM, AND DAVID
if there
@ni]stable
A is inertia-preseruing
in AD = in D. We matrices
TIN-YAU
AD + DAT is positive
stable matrices
arise in problems
UHLIG,
D-stable,
ecology,
chemistry,
is not true, as shown by the following
EXAMPLE 1.
Let
A=[;
Here
A is D-stable
[6] but not inertia
[2], that
diagonal
because,
e.g. [2,
matrix
clearly,
D,
inertia
example.
-5;].
preserving,
since for D = diag{ - 1,3, - 1) the
AD is also stable.
A subclass
of the
[l]: matrices
matrices
D-stable
matrices
A such that
only if D is positive. Manus
1
diagonal
e.g.
and economics,
invertible
This is of interest
The converse
matrix
exists a positive It is known,
and diagonally stable matrices
if for every
matrices.
CARLSON
is the
class
AD is stable,
where
Again, it is clear that inertia
of
Arrow-McManus
D-stable
D is a diagonal matrix,
preserving
matrices
if and
are Arrow-Mc-
D-stable.
Example D-stable.
1 is also an example
Observe
A real matrix preserving
D-stable.
D-stable A is
invertible)
and diagonally
strongly
matrix
inertia
semistable,
preserving
submatrices
that
real
but not strongly inertia preserving.
diagonal
A is strongly
are inertia preserving.
The matrix
is inertia preserving
but not inertia preserving.
if for every
D, in AD = in D. Observe
if and only if all its principal
EXAMPLE 3.
[7]. In fact, we show
Let
A is Arrow-McManus
necessarily
matrix which is not Arrow-McManus
semistable
Let A be a diagonally semistable matrix. Then A is D-stable if and only
THEOREM 1.
if it is Arrow-McManus EXAMPLE 2.
of a D-stable
that it is not diagonally
(not
inertia
AUBURN 1990 CONFERENCE
737
ON MATRIX THEORY
An important class of inertia preserving (and even strongly matrices is the class of diagonally stable matrices.
inertia
preserving)
A diagonally stable matrix is strongly inertia preserving.
THEOREMS.
Clearly, if A is inertia preserving, then io( A) = 0. The following theorem terizes the real square matrices A such that io( A) = 0. Recall [7] that B,(A) denotes the cone B,,(A)
(a) (b) (c) (d)
= (BEPSD[(BA)~~=O,~=I,...,~)
The folhving
THEOREMS.
charac-
properties
of A E R”, n are equivalent:
io( A) = 0.
BEPSD, BA+ATB=O * B=O. BcPSD,BA+ATB=0,rankB<2 t+ B=O. B EB,,( A), BA + ATB = 0, o B = 0. (e) BEB,,(A), BA+ATB=0,rankB<2 o B=O. For matrices which have no pure imaginary eigenvalues THEOREM
4.
one has
Suppose io( A) = 0. Then
(a) in A = in AD for every positioe diagonal matrix D if and only if
(b) io( AD) = 0 fw every positive diagonal matrix D. Given that a matrix A is stable, we obtain D-stability as a corollary of Theorems 3 and 4. Let A be a real stable matrix.
COROLLARYl.
every B E Bo( A) of rank
4.
a simple
characterization
of
Then A is D-stable ifand only iffor
< 2 and for every positive diagonal matrix D, BAD+DATB=O
EXAMPLE
here
o
B=O.
The matrix
is stable. To show that it is D-stable we observe that B E B,( A) if and only if it is of the form a 0 ob’
B=
-4c
0
-4c 0, C
a, b, c > 0, I
a > 16~.
738
FRANK
UHLIG,
TIN-YAU
TAM, AND DAVID
CARLSON
In this case
4c-
a
16c
a0 0
and if BAD is skew-symmetric
for an invertible
are equal to zero, so BA = 0, which implies Since
inertia
matrices,
preserving
matrices
it is natural to ask whether
preserving.
Note that Example
two classes coincide
THEOREM
are
matrix
D-stable
and include
diagonally
this question
acyclic
c, a, and b
the diagonally
semistable
matrices
in the negative.
stable
are inertia
However,
the
matrices.
Let A be an acyclic irreducible
5.
D, then necessarily
that B = 0.
D-stable
2 answers
for irreducible
I I
matrix.
Then A is D-stable if and only
ifA is inertia presweruing. If the irreducible
2.
COROLLARY preserving
if and only if it
The following property
THEOREM
components
of A are acyclic,
of diagonally
semistable
matrices
is of great importance.
Let A E R”,” be a diagonally semistable matrix,
6.
invertible diagonal matrix.
then A is inertia
is D-stable.
and let F be an
The following are equivalent:
(a) in AF = in F (b) io( AF) = 0. (c) BAF + FATB # 0 for every nonzero B E B,( A) s.t. rank B < 2. (d) BAF + FATB # 0 for every nonzero B E B,( A). Observe general,
that
the
implication
as shown by Example
COROLLARY
(b),(c),(d)
i* (a) in Theorem
Let A be a diagonally semistable
3.
6 does
not hold
in
1.
matrix.
The following are equiva-
lent: (a) A is inertia preserving. (b) io( AF) = 0 for every real invertible diagonal matrix F. (c)
BAF + FATB # 0 for
every
nonzero
BE Bo( A) such
that rank B Q 2, and fw
every real diagonal invertible matrix F. (d) BAF + FATB f 0 fat every nonzero B E B,( A), and for every real diagonal invertible matrix F. We conclude
QUESTIONS.
by asking the following
questions:
Is every strongly inertia preserving
matrix diagonally
stable?
AUBURN 1990 CONFERENCE QUESTION 2.
Is every irreducible
The proofs and additional
739
ON MATRIX THEORY inertia preserving
matrix diagonally semistable?
examples will appear in [3].
We wish to thank Professor Daniel Hershkowitz for many suggestions which improved the paper.
REFERENCES K. J. Arrow and M. McManus, A note on dynamic stability, Econometrica 26 (1958). G. P. Barker, A. Berman, and R. J. Plemmons, Positive diagonal solutions to the Lyapunov equations, Linear and Multilinear Algebra 5:249-256 (1978). A. Berman and D. Shasha, Inertia preserving matrices, SZAM /. of Matrix Anal. Appl., to appear. B. L. Clarke, D-stability and chemical reaction networks, presented at the Combinatorial Matrix Analysis Conference, Victoria, 1987. G. W. Cross, Three types of matrix stability, Linear Algebra Appl. 20:253-263 (1978). D. J. HartRel, Concerning the interior of the D-stable matrices, Linear Algebra Appl. 30:201-207 (1980). D. Hershkowitz and D. Shasha, Cones of real positive semidefinite matrices associated with matrix stability, Linear and Mu&linear Algebra 23:165-181 (1988). J. F. B. M. Kraaijevanger, A characterization of Lyapunov diagonal stability using Hadamard products, Linear Algebra Appl., submitted for publication. J. F. B. M. Kraaijevanger and J. Schneid, On the unique solvability of the Runge-Kutta equations, Numer. Math., submitted for publication.
CONJUGATE-SUBSPACE DECOMPOSITION AND ITS APPLICATION SOLVING LINEAR SYSTEMS WITH MANY RIGHT-HAND SIDES by MEI-QIN
IN
CHEN31
The inspiration for the conjugate subspace decomposition (CSD) has come from the updated conjugate subspace method for solving unconstrained minimization problems whose objective functions are twice differentiable, as first introduced in [5]. We assume throughout this paper that A is an n x n symmetric positive definite matrix. The CSD can be described as follows [3].
31Department of Mathematics/Computer
Science,
The Citadel, Charleston,
SC 29409.
740
FRANK THEOREM 1.
UHLIG,
TIN-YAU
Let R” be a conjugate sum of T,,
TAM, AND DAVID
CARLSON
. . , T,,, with respect to A, that is,
(1) R” = T, + *.. +T,; (2)
rfArj
= 0 for xi E Ti, rj E Tj, and i # j.
+ * * . +I(~)
Then x* = 8) XE ‘1;:. The
following
equations
is an algorithm
based
on the CSD
for solving
systems
of linear
Ax = 6. (CSD).
ALGORITHM matrices
solves Ax = b, x E R”, where each xci) E Ti solves Ax = b,
Let R” be the conjugate
sum of T, . . . T,, and Zi be the basis T,, respectively. Let nj = rank Ti, where n, + .- . + n, = n. x* of Ax = b can be computed as follows:
of the subspaces
Then the solution (1) Compute
bj = Zfb, i = 1,. . . , m.
(2) Solve Z:AZ, fci) = bi for ??ci)E R”i, i = 1, . . . , m. I ,..., m. (3) Evaluate x (i)=Z,&i),i= (4) Set r* = x(l) +
*. * +x(@.
The
allows
CSD
strategy
ni-dimensional problems.
subproblems
Some questions,
scale problems
however,
ment in efficiency?
subspaces
a class of problems
x(t)
numbers. when
need to be answered:
The
the
b(t)
(CC)
(BCG)
multiprocessor
In general,
time; for example,
the vector
this case, the BCG algorithms
algorithms
sequential
in
definite, of
P)
the right-hand-side
vector
t, and S is a set of discrete
is suitable
for solving
and without
an explicit
[7] are adaptable
for all
b
real
(P), especially form.
If the
t in S, then the
and work efficiently
on a
the b(t) may not be available at the same
are no longer adaptable.
If the vectors
then the Lanczos-Galerkin
r(tj) for j < i. In b(t) do not change
projection
error bound of the approximation
If this is not the case, then computing
such problems.
the solution
at step (2) to find each solution
It has a high parallelism
of the CSD,
of the CG.
t E S,
procedure
of the solution is
of (P) is completely
t. The algorithm combining the CSD strategy with the CG algorithm, that
is, applying the CG algorithm because
subspaces?
b(tj) may depend on previous solutions
in [9], and its theoretical
given in [lo].
(1) For what type of large
simultaneously
however,
too much from one solution to another, is suggested
method sparse,
are available
block-conjugate-gradient machine.
positive
functions
A in (P) is large,
vectors
m
with less cost and with great improve-
for each
= b(t)
conjugate-gradient
right-hand-side
by solving
for solving large scale
of the form
that are real-valued
matrix
problem
(2) For a given linear system of equations,
be chosen
E R”, A E R “xn is symmetric
has components
n-dimensional
(3) What is the cost of forming these conjugate
Ax(t) where
an
and has great potential
should the CSD be adapted?
how should the conjugate Consider
us to solve
in parallel
and it does not require
in computation the explicit
x-ci), is suitable for solving for solving
(P) for each
form of the matrix
A, because
t
AUBURN
1990 CONFERENCE
ON MATRIX
Notice that the basis matrices set S, the cost of forming So it is important efficiently. the
The performance
spectrum
unfavorable
of
distributions methods matrix
A in the
spectral
well-separated,
Zi need to be formed only once. With many t in the
Ti can be compensated
to choose
of the CG algorithm presence
is very sensitive
of roundoff
errors
In practice,
can be found, for instance, parameters
[l,
the spectrum
when the relaxed
w close
of solving (P).
solves each subproblem to the distribution
6, 8, 9, 11,
for the CC is when the spectrum
eigenvalues.
with varied
by the overall efficiency
Ti’s such that the CG algorithm
distribution
large
741
THEORY
has a few distinct,
of A in (P) with such
(block) incomplete
to 1 are adapted
of
121. One
Cholesky
as preconditioners
to the
A which results from discretizing
-Au=ffor(r,y)EOand
u=Ofor(x,y)~afi,
by linear
finite
examples
with different
element
approximations choices
where
over a uniform
the relation
between
following are two theorems projections approximate
on the
eigenspace
subspaces
the spectrum
triangulation.
Some
one
of the
for a given A, it is necessary
of A and its projections
[3] which describe
Ti’s when
isosceles
x (0,l)
of w are given in [2].
In order to form a set of proper conjugate investigate
0 = (0,l)
the relation subspaces
of the spectrum
contains
to
on the Ti’s. The of A and its
an eigenspace
or an
of A.
THEOREM 2. Let (X,,qJ,i = l,...,m, be the eigenvalue-eigenvector pairs of A, and let Q = [ql. . . . , q,,J. If span Q E Ti f or some i, then span Q I ?; for j + i. lf Zj’Zj = I,,, for each j, and (hi, qi) are extreme eigenpairs of A, then X,, . , . , A,,,are not in the spectrum of A on Tj for j f i.If in addition ZfZj = 0, then the former statement is aLso true for intermediate eigenpairs.
. . ,4,1, w/m-e 1 C ml3 m2
THEOREM 3. Let iT, = [&, . . . , g,,,], & = [q_,,+l,. Q n, and let pk = i$ A& /&& for each k be such that
I pk
-
‘k
I = “$
I Pk
-
$1,
k=
l,...,m,,
1~
k=
and I &t-k
If span[&,
-
h-k
I =
jF;yk
I /hi-k
-
h-k
l,...m,.
G2] E Ti for some i, then fm j # i,
[
k=.,$,,,,,
(:)‘1’:.
742
FRANK
Furthermore, in the inter&
UHLIG,
TIN-YAU
TAM, AND DAVID
CARLSON
if ZiZj = l,,, fm each j, then the spectrum of A on q fw j # i is contained [a, b], where
a= &,+1 -
k$l (xm,+1 - Ak)Ck(Ek)>
where the functions ck are such that
ck(o) = 0
Theorem
2 and Theorem
the CC algorithm, of the conjugate
to (i) choose a subspace
of A whose corresponding proper algorithm
eigenvalues
Ax(t)
ri are normalized = b(t), for some
(1) The vectors -
appear in the spectrum
of A on the
conjugate
vectors
we need
eigenspace
to the CG, so that the CG
subspaces
efficiently;
(ii) find a
on T1.
to form such a subspace
residual
more efficient,
or an approximate
are not favorable
to solve the subproblem
to
of A on one
an eigenspace
of A on the other
One of the natural choices where
which are not favorable
in the spectrum
In order to make the CG algorithm
T, which contains
solves the subproblems
= 0
are embedded
then they no longer
subspaces.
&k)
&k
3 show that if the eigenvalues
or their approximations subspaces,
other conjugate
ck(
lim E-o
and
generated
T, is to let Z, = [?,,,
by the CG algorithm
. , Pm],
for solving
t E S. There are four advantages to such a natural choice:
ri can be formed throughout
the computations
by their recurrence
relation. (2) The quantities can be computed
ci used to estimate
explicitly
(3) Since T1 = K,_,,
the Krylov subspace
m such that T, contains well-separated
extreme
(4) The matrix performance
an eigenspace
3
of dimension
or its approximation
m, there exists an integer corresponding
to those
eigenvalues.
Zf AZ, is a tridiagonal
is not affected
of A, such as the direct
algorithms
whose
method,
can be
Z: AZ, f(l) = b,.
complement
T, = span Z, = null[( AZ,)‘].
matrix, and many efficient
by the spectrum
adapted to solve the subproblem To form a conjugate
the lower and upper bounds in Theorem
or can be easily estimated.
of subspace
A Householder
T,
with respect
orthogonalization
to A, we may let
procedure
can be ap-
AUBURN 1990 CONFERENCE
ON MATRIX THEORY
743
plied here to form an orthonormal basis which generates the subspace Ts, and the spectrum of A on T, can be estimated by Theorem 3. This decomposition can be carried on again on the subspaces to that a set of conjugate subspaces are formed successively. Observe that if Ta is chosen as an orthogonal complement of Tr, then its basis matrix 2, can be formed as 1 - ZrZi without any extra cost. With this choice of ‘I’,, the solution of(P) is no longer simply the sum of the subsolutions on Tr and Ta, but it can be computed from those subsolutions by using the Sherman-Morrison-Woodbury formula. The complexity and the performance of the algorithm with these choices, and the scheme used to determine the dimension of Tr, are discussed with details in [4].
REFERENCES 1
2
3
4
5
6 7 8 9 10 11 12
0. Axelsson and G. Lindskog, On the Eigenvalue Distribution of a Class of Preconditioning Methods, Group Report 3, Dept. of Computer Sciences, Chalmers Univ. of Technology, Goteborg, Sweden, 1985. 0. Axelsson and G. Lindskog, On the Rate of Convergence of the Preconditioned Conjugate Gradient Methods, Group Report 5, Dept. of Computer Sciences, Chalmers Univ. of Technology, Goteborg, Sweden, 1986. M.-Q. Chen, The Updated Conjugate Subspace Method in Optimization and in Solving Linear Systems of Equations, Ph.D. Thesis, Dept. of Mathematics, Univ. of Illinois at Champaign-Urbana, 1989. M.-Q. Chen and A. Sameh, The Conjugate Subspaces Decomposition and Its Application in Solving Linear Systems of Equations with Many Right-Hand Sides, Tech. Report 1024, Center for Supercomputing Research and Development, Champaign-Urbana, Aug. 1990. S.-P. Han, Optimization by updated conjugate subspaces, in Numerical Analysis (D. F. Griffiths and G. A. Watson, Eds.), Pitman Res. Notes Math. Ser., 1986, pp. 82-97. A. Jennings, Influence of the eigenvahre spectrum on the convergence rate of the conjugate gradient method, J. Inst. Math. AppZ. 20:67-72 (1977). D. P. O’Leary, The block conjugate gradient algorithm and related methods, Linear Algebra Appl. 29:243-322 (1980). C. C. Paige, The Computation of Eigenvalues and Eigenvectors of Very Large Sparse Matrices, Ph.D. Thesis, Univ. of London, 1971. B. N. Parlett, A new look at the Lanczos algorithm for solving symmetric systems of linear equations, Linear Algebra Appl. 29:323-346 (1980). J. &ad, On the Lanczos method for solving symmetric linear systems with several right-hand sides, Math. Cump. 48(178):651-662 (Apr. 1987). A. van der Sluis and H. A. Van der Vorst, The rate of convergence of conjugate gradients, Numer. Math. 48:543-560 (1986). A. van der Sluis and H. A. Van der Vorst, The convergence behavior of Ritz values in the presence of close eigenvahres, Linear Algebra Appl. 88/89:651-694 (1987).
744 INVERSES
FRANK UHLIG, TIN-YAU TAM, AND DAVID CARLSON OF MATRICES
ARISING
FROM
DIFFERENCE
OPERATORS
by SUI SUN CHENG3*
Explicit inverses and inversion algorithms for square matrices have been a major concern since the early days of matrix theory. As the areas of application of matrix theory began to broaden, band and Toeplitz matrices [2] found their way into several methods of numerical analysis and approximation theory. For example, they would arise when using finite difference methods for differential equations, when using polynomial splines, and in studying discrete random processes [3] and statistics [4]. When using finite difference methods, for instance, it is desirable to find accurate error bounds. Explicit inverses or properties such as bounds or asymptotic behavior of the elements of the inverses are needed for this purpose. A common way to obtain this information is to observe that each column of the inverse of, say, a Toeplitz matrix satisfies a linear recurrence equation wth constant coefficients. This recurrence equation can then be solved, at least in theory, and the solution expressed as a linear combination of powers of roots of the characteristic equation associated with the recurrence equation (see e.g. [5, Chapter 41). Thus, the properties of the elements of the inverse can be deduced by this process. As an example, consider the well-known tridiagonal matrix A = (u~~)“~,,, where aii = - 2, aij = 1 if ( i - j 1 = 1, and aij = 0 otherwise. If we denote the jth column vector of the inverse A- ’ by col( x(l), x(2), . . , x(n)), then the components of this vector satisfy
x(k -
1) -
2x(k)
+ x(k +
1) = 6kj,
k=
1,2 ,...)
It,
where we have defined x(O) = 0 and x(n + 1) = 0, and Likj= 1 if i = j and iikj = 0 otherwise. It is convenient to employ the forward difference operator A, defined by Au(k) = u(k + 1) - u(k), to write the above equations as
A%(k
-
1) = i_ikj,
k=1,2
,...,
n,
x(o) = 0 = %(rt + 1). Taking our cue from the theory of Green’s functions in differential would guess that the solution of the above problem is of the form
equations,
we
x(k)=a+bk+(k-j),,
32Department
of Mathematics,
work was funded by the National
Tsing Hua University, Research
Council
Taiwan,
of the Republic
Republic of China.
of China.
This
AUBURN 1990 CONFERENCE
745
ON MATRIX THEORY
where (Y+= Q if cr > 0 and LY+= 0 if (Y< 0. Then it is easily calculated that a = 0 and b = - (n + 1 - j)/(n + 1). By symmetry considerations, we would also guess that
x(k) = c + d(n + 1 - d) + (j - k)+ and deduce that c = 0 and d = - j/(n + 1). By means of the definitions and (j - k) +, we see that x(k) is also given by
of (k - j),
k(n+l-j)
x(k)
=
I
n+l
’
k
j
_j(n+l-k), n+l
This example motivates a generalization for constructing explicit inverses of matrices for which the inner product of its kth row with the jth column of its inverse can be written in the form A”%(k - t) = Skj,
(1)
where m + 1 is the band width and 1 Q t Q m - 1. The details of this generalization will appear elsewhere [l] and will not be repeated here. However, we shall quote the following theorem, which is central to the derivations of explicit inverses. THEOREM. Let 2 4 m < n, 1 4 t 6 m - 1, and 1
k=
. . . . -1,&l
G n. Let P,,,_l be the set of
with degree less than or equal to m - 1. Then x(k), ,... by
+)
p(k)
z
+
defined fw
(k-‘(; t-l;/:“-1), P~P?l-,
or
+)
=
+)
+
(-l)“(j -;m’_ml-,l + #?-“,
satisfies Equation (l), wher-e fw c@ = cY(a - 1) *. 9 (a -p+l) j3 = 0.
any integers if
a>0
9EP,.-,,
(Y and & @) = 0 if (Y < 0 or /3 < 0,
and
/3>0,
and c@)=l
ifa>0
and
We remark that even though explicit inverses can be constructed in principle by the above theorem, we have not derived subsequent properties of the inverses. It will be of interest to find norms or bounds or asymptotic behavior of the elements of the inverses by means of the method mentioned above.
746
FRANK UHLIG, TIN-YAU TAM, AND DAVID CARLSON
REFERENCES 1
S. S. Cheng
and L. Y. Hsieh, Inverses
of matrices
arising from difference
operators,
Utilitas Math., to appear. 2
D. S. Meek, A survey of the results on the inverses of Toeplitz Proceeding of the Conference RIMS,
Implementations, 3
B. N. Mukherjee matrices
W. F. Trench,
5
K. S. Miller,
12:515-522
definite
Toeplitz (1988).
An algorithm for the inversion of finite Toeplitz matrices,
J. SIAM
to the Cakx~us of Finite Differences
An Introduction
and Di&jerential
Dover, New York, 1966.
MINIMAL RANK AND MAXIMAL FOR CERTAIN BAND MATRICES by JEROME
RANK HERMITIAN
COMPLETIONS
DANCIS33
In the last decade Hermitian
of positive
Linear Algebra Appl. 102:21 l-240
applications,
(1964).
Equations,
specified
in
and the
Kyoto Univ., Kyoto, Japan, July, 1982.
and S. S. Maiti, On some properties
and their possible
4
and band matrices,
on Standard Algorithms jw Linear Computation
Hermitian
a popular
matrix either
problem
has been
to a Hermitian
to try to “complete”
matrix with prechosen
a partial
inertia
or to a
matrix with the maximum or the minimum possible rank. The purpose of this
paper is to present band matrices
the standard
together
method
for constructing
Hermitian
completions
of
with some results on minimal and maximal rank Hermitian
completions. The
DEFINITION.
{ rr( H), v(H), 6(H)}
inertia
consisting
ues of H. We let r(H), Our starting
THEOREM 1 [I].
n x n Hermit&
a
Hermitian
matrix
H
of positive,
is
negative,
a
is our generalization
of Poincare’s
triple
In H =
and zero eigenval-
Y(H), and 6(H) denote the three coordinates
point
interlacing theorem
of
of the numbers
of In H.
inequalities
and Cauchy’s
(notation: 0’ = 0 E Cr):
Let R, be the leading principal
matrix
H.
(n -
r) x (n - r) submatrix
of an
We set A = Dim Ker 23, - Dim[(Ker R,) @ Or] fl Ker H.
Then (a) 7r( H) 2 T( R,) + A and (b) r(H) Clearly,
> r(R,)
+ A.
the inertia
of every
Hermitian
completion
of a (band) matrix
must be
consistent with this theorem.
33Department of Mathematics, University of Maryland, College Park, MD 20742.
AUBURN 1990 CONFERENCE
ON MATRIX THEORY
747
BORDERED-MATRIXHYPOTHESES. Let H, be an (r - 2) x (r - 2) Hermitian matrix. Let o and w be vectors in C’-‘, and let a and b be real numbers. Let H(z) be the bordered matrix
H(Z)
=
a v
::
.z* w ,
Iz
w*
b 1
and let
and
H, =
H(z) is called a one-step compkiun of H(0). The one-step-completion problem for these bordered matrices is usually the inductive step in the proofs of completion theorems. We used Theorem 1 as we classified all the possible kernels of bordered matrices in Lemmas 3.3-3.9 of [2]. A sample of the “common threads” of these lemmas is Theorem 2. THEOREM 2 (Theorem 1.2 of [2]). Given the bordered-matrix hypotheses. Suppose that the nullities are non-decreasing, namely,
6(H2)
and h(h) 6
B(h)
(that is, the s&matrix does not have a larger nullity). Then there is a number z. such that there is a preservation of positivity and negativity, namely,
“(H(Q)) =M~{+++G)j
and +(G))
= M~{~(H+OS))
and again the nullities are non-decreasing, namely,
6(HI) g 6(H(G)) (that is, the new s&matrices
and 6( Hs) 6
h(H( ~0))
do not have larger nullities).
DEFINITION. A matrix with all zeros off the main diagonal and the first m pairs of superdiagonals is called an m-band matrix, that is, R = (rjk) and rjk= 0 for all (k - j ( > m. An n X n Hermitian matrix F = (f$) is a compktion of an m-band matrix fl if fjk = rjk for all 1k - j ) < m. The maximal Hermitian (m + 1) x (m + 1) submatrices R,, R,, . . . , R,_, within an m-band n x n matrix R are Ri = ( rjk Ij,k = i, i + 1, . . . , m + i). The almost maximal Hermitian m x m submatrices of R are RT = (rjk (j,k = i, i + 1, ...,m + i - 1). RT, R;, . . . , &,,+I
748
FRANK UHLIG, TIN-YAU TAM, AND DAVID CARLSON A simple diagonal completion R’ of an m-band Hermitian
DEFINITION. an m + l-band
Hermitian
of an m-band
completion
(N - 1)st (successive)
Hermitian
as the s&matrices
hypotheses.
Also,
completion
matrix
An
N th (successioe)
R is a simple
H,,
H,,
and
H,,
completion
Ri and Ri+l
of an
and their “com-
of the bordered-matrix
respectively,
the ith maximal
the maximal
almost maximal s&matrices
diagonal
R is
diagonal
RT of a band matrix R always overlap in the same
H( .zo) becomes
R’. In addition,
matrix
simple
of R.
Each pair of maximal submatrices
almost maximal submatrix
manner
of R.
simple diagonal completion
OBSERVATION 3. mon”
completion
of the simple
submatrix
submatrices
diagonal
of a band matrix R become
the
of its simple diagonal completion R’ (with the same ordering
from the upper left corner). STANDARD METHOD (For matrices). R’, then
Complete complete
diagonal completion completion
R’ to R” =
completions will construct Theorem
a Hermitian 4 below
of our
and negativity
completion
3, and the Standard
Method
rank Hermitian submatrix
completion
and 6(Ri+1)
property.
will establish completion
of a Hermitian
= Max{*(Ri),i
method”
As an example, the next theorem.
result.
It says that
band matrix
R is not
then there is a positivity
Given a Hermitian
RI, R,,
m-band matrix R with
. . , R,_, and R:, RX,. . , R*,-,+l.
namely, foreach
i=1,2
,...,
n-m-l.
F of R such that there is preservation
= 1,2 ,...,
= Max(v( A,), i = 1,2,. (We announced
of
completions
the “standard
namely, z(F)
kernels
on one-step
of R.
2 6(Ry)
completion
possible
By using these one-step
of its pair of almost maximal submatrices,
Then there is a Hermitian
simple diagonal
theorems
completions,
with the desirable
Suppose that the nullities are nondecreasing,
and negativity,
of the
properties.
THEOREM 4 (Minimal rank completions).
> 6( Ry)
classification
simple diagonal
is a minimal
preserving
A to
completions.
desirable
maximal and almost maximal submatrices
6(R,)
3, each successive
one-step
of [Z]), one may obtain
the nullity of each maximal
less than the nullities
n x n
complete
F is an (n - m + 1)st simple
3.3-3.9
of successive
2, Observation
Theorem whenever
threads”
(Lemmas
which propagate
in the construction
of m-band
namely,
is the method used in [3-71.
“common
matrices
completions
(R’)‘. . . to F, where
by a set of independent
This standard method bordered
Hermitian
simple diagonal completions;
of R. After noting Observation
is achieved
By finding
constructing
successive
n - m}
and
V(F)
. . , n - m).
this result at the ILAS meeting
in Provo, Utah, 1989.)
of positivity
AUBURN 1990 CONFERENCE
749
ON MATRIX THEORY
Of course, the inertia of a Hermitian completion F of a partial Hermitian n X n matrix R must be consistent with Poincare’s inequalities and Cauchy’s interlacing theorem, that is, for each maxim+ specified principal submatrix R, of R, r(F) ) *( RJ and V(F) ) v( Ri). Therefore Theorem 4 provides the minimal possible rank among all possible Hermitian completions. Dym and Gohberg presented the standard method in [5] as they showed that, when all the maximal submatrices of a Hermitian band matrix R are positive definite, then R has a Hermitian completion F which is also positive definite, such that F-’ is also a band matrix. They report that this result has connections with signal processing and system theory. Johnson and Rodman have shown (in [7]) that when all the maximal submatrices of a Hermitian band matrix R (or even more generally a matrix with a “chordal” graph) are invertible, then R has an invertible Hermitian completion. Ellis, Gohberg, and Lay have shown (in [S]) that when all the maximal submatrices and all the almost maximal Hermitian submatrices of R are invertible, then R has an invertible Hermitian completion F for which F-’ is also an m-band matrix. THEOREM 5 [3]. < n - m, suppose
LetRbeanm-bandn~nmutrix. that R,
Furanintegerr,m+5<2r
or R’f is invertible.
Then R has an invertible
Hermitian
completion. COUNTEREXAMPLE 6. An example of a l-band 3 x 3 matrix, with an invertible maximal Hermitian submatrix, which does not have an invertible completion is
1
010 1 01
0 0.
I
Theorem 7 is a generalization of Theorem 5, but its Hermitian or may not be the one with the maximal possible rank.
completion
F may
THEOREM 7 [3]. Let R be a Hermitian m-band n X n matrix. Suppose that m 6( R,) + 5 Q 2 r < n - m for some integer r. Then R has a Hermitian completion F with S(F) Q 6(R,).
DEFINITION. Given an m-band n x n matrix R = (rij), its maximal full-column is the unique specijied n x (2m + 2 - n) submattix
submatrix
M=
(rij(i=
1,2,...,
r
and
j=n-m,n-m+l,...,
m+l).
REMARK. Since the columns of M will also be columns of any completion Ker F > On-m-1
e Ker M e O”-m-1
and
RankF,<2(n-m-l)+RankM
F of R,
750
FRANK UHLIG,
for any completion matrix,
TIN-YAU TAM, AND DAVID CARLSON
F of R. In the trivial case, when 2m + 2 < n and M is an empty
we set Ker M = 0 and Rank M = 0.
provided by Theorem
Therefore
the
Hermitian
completion
8 has the maximal possible rank among all (including non-Hermi-
tian) completions. 8 (A maximal rank, minimal kernel completion
THEOREM
n x n Hermitian matrix R with maximal s&matrices IRank Ri - Rank Ri+l
1< 1
forall
R,, R,,
[4]).
1,2 ,...,
i=
Given an m-band
, R,_,.
.
Suppose that
n-m-
1.
Then for almost all Hermitian completions F of R, Ker F = O”-m-1
@ Ker M @ On-m-1
Rank F = 2( n - m - 1) + Rank M,
and
where M is the full-column maximal submatrix
of R. if R is also a real symmetric matrix, then the conclu.sions are valid for all but possibly a finite number of the real symmetric completions. Furthermore,
Curiously, Theorem
REMARK. demonstrated
8 is not valid for non-Hermitian
band matrices,
as is
by this l-band 3 x 3 matrix:
F( n, y) = 1 Here Rank F( x, y) = 2 # 3 = 2(3 -
1 -
0 0
1 0
x 0
Y
1
0
. i
1) + Rank M for all values of x and y, even
though Rank R, = Rank R, for the only maximal specified submatrices.
REFERENCES Jerome
Dancis,
operators,
On the inertias
of symmetric
Linear Algebra Appl. 105:67-75
Jerome
Dancis, Bordered
Jerome
Dancis,
matrices,
Invertible
matrices
and bounded
Linear Algebra Appl. 128:117-132
completions
self-adjoint
(1988).
of band matrices,
(1990).
Linear Algebra Appl., to
appear. Jerome Dancis, Minimal Rank and Maximal Rank Hermitian Completions Certain Band Matrices, Technical Report Tr 89-58, Univ. of Maryland, 1989. Harry Dym and Israel Gohberg,
Extensions
of band matrices
for
with band inverses,
(1981).
Linear Algebra Appl. 36:1-24 Robert L. Ellis, Israel Gohberg,
and David C. Lay, Invertible self-adjoint extensions
of band matrices and their entropy,
SIAM J. Algebraic
Discrete Methods 8:483-500
(1987). Charles
R. Johnson
partial Hermitean
and Leiba
matrices,
Rodman,
Inertial
Linear and Mu&linear
possibilities
for completions
Algebra 16:179-195
(1984).
of
AUBURN
ON CHARACTERIZATIONS OPERATORS by
THEORY
~~~~C~NFERENCEONMATRIX
OF THE SPECTRAL RADIUS OF POSITIVE
SHMUEL FRIEDLAND Let B be a Banach space over the real numbers
the Banach space of all real-valued pointed
(K fl - K = (0))
function&
with
cone.
respect
to
K.
bounded Denote
then let
x ( y iff
B = K - K and bounded
K* f (0).
in norm, then
linear operator.
Denote
Note
let
y 2 x and
Furthermore,
interior,
called
positive
if
Collatz-Wielandt
AK C K. Assume
v(A)
sets associated
-e
: B + B be a bounded
the local spectral
radius of A at x.
linear operator
A is a positive
operator,
A : B + B is
and define
= (o:3r>O,
Ax
X1(A)
= {a:3x%+O,
Ar
“(A)
= {w:3x>O,
Ax>wr},
$(A)
= (w:3r+O,
Ax>wx}
the upper and lower Collatz-Wielandt =inf{a)O:Ar
R(A,x)
the
with A [2]:
“(A)
For xcK
that
Q x < e is
= mimi)lhl.
p( A, r). A bounded that
A
by K,.
implies
of A. Set
For x E B let p( A, x) = lim sup I( A’% ((‘lrn denote ,tal
K, # 0
the segment
B* = K* - K*. See [7j or [S]. Let
linear
y - r E K and
and denote its interior
if for eEK,
by a( A) the spectrum
T c B set V( A, T) = inf,..r
by B*
K C B be a closed
y > x iff
that the assumption
P(A) = pan, I XI,
For
]] I]. Denote
Let
by K* C B* the cone of nonnegative
As usual,
y - XEK,.
with the norm
linear function&.
y - r E K \ (0). Assume that K has a nonempty We
751
numbers
are defined as follows:
r( A, x) = sup{w
> 0: Ax > UK}.
(1)
Note that R( A, r) = 00 if no u exists such that An < UT. Clearly,
sup Qr( A) = :“,“or( A, x) G sup Q( A) = supr( X>O infX(A)
In the case of B = numbers
= mf,R(A,x)
R”, K = R;, and A an irreducible
A, x),
\
= I=f,R(A,x).
nonnegative
matrix, all the above
are equal to p(A). This is the classical result due to Wielandt.
any finite-dimensional
34Department
B with a closed
of Mathematics,
pointed
spanning
cone
I
It also holds for
K and a positive
University ofIllinois at Chicago, Chicago, Illinois 60680.
752
FRANK UHLIG,
irreducible
operator
A. However,
TIN-YAU TAM, AND DAVID
if A is not irreducible
have to be equal even in the finite-dimensional following theorem
characterizes
local spectral
do not
[6] and [9]. The
in terms of the spectral
radius
radii of A and A* [5]:
Let B be a real Banach space with a closed pointed cone K. Assume interior and B* = K * - K*. Let A : B + B be a
THEOREM 1. bounded
the above four numbers See for example
the above four numbers
and the minimal distinguished
furthermore
case.
CARLSON
that K has a nonempty
linear operator which leaves K inuariunt.
Then
(3) ji$R(
A, x) = v( A, K),
sup r( A, x) = r~( A*, K*),
(5)
x*0
supr(A,x)
x>o Zj p(A) i.s in Recall pointed;
of A, then
the point spectrum
that
a finite-dimensional
Let C be a C*-algebra. of the form aa*. Let
Denote
K
is generating
B Theorem
(B = K - K)
of K, i.e.,
iff
is
K*
1 is due to [9].
by K the cone of all self-adjoint
be the interior
K,
equality holds in (6).
cone
e.g. [l]. For a finite-dimensional
(6)
(positive)
elements
K, is the set of invertible
elements
in K. THEOREM 2 [5].
Let C be a C*-algebra.
positioe linear operator p(A)
Assume
that A : C -+ C is a bounded
(AK C K). Then
= ,i{
u( A*, K*)
@‘Ax),
= sup v( &4x).
0 The above theorem tion.
Indeed,
product
on
self-adjoint
can be considered
C: (a, b) = +(ab*),
QC Endz(C).
as an extension
if C is a finite-dimensional
positive
elements
where
of C, e.g.
Note that the subalgebra
$? is a semisimple
algebra.
(7)
EK”
C* algebra, 4
is a positive
[8]. Then
functional
characteriza-
on the
C can be viewed
@? is invariant
More precisely,
of Wielandt’s
then one can define
vector
cone
of
as a subalgebra
under the involution
the underlying
an inner
space
*. Hence,
C splits to a
direct sum VI + * * * + Uk where each Vi is an irreducible invariant subspace under the action of ‘iR. It then follows that the restriction of V to uj is isomorphic to M,,(C), where mi = dim Vi. Thus, any finite-dimensional
C*-algebra
c = Mm,(C) + *. * +%f,,(c) .
is isomorphic
to
AUBURN
1990 CONFERENCE
ON MATRIX
753
THEORY
Here,
a=
( a,,...,
Q),
aiEMmi(C),
@I,..
i = l,...,
.>a#$.
.>bk) = (a$,,..
Thus, if A : C + C is a positive operator elements,
ly,ykP(x;‘( Ax)i),
a;)>
.dQbk)>
to the cone of self-adjoint
v( x-'Ax)
. .
(xl,. . ., xk),
Ax = ((Ax)~,..
Note that C is isomorphic C is a commutative i = 1,.
with respect
a* ( r,...,
That
under the pointwise
is, in (8) we have
characterization
C = M,(C)
for a nonnegative
and a positive
Consult
irreducible
operator
[4] for other characterizations
with respect
to general
irreducible
Ax)i),
i = 1,. . .,k.
xi = aia’,
.,(Ax)~),
to Ck as a C*-algebra
C*-algebra.
mh,v( x;l(
=
the
matrix
(9)
multiplication
equalities
. . , k. As p(a) = Y(U) = a for 0 < a E M,(C) = C, we deduce
Wiehmdt
positive
we deduce
P( x-‘Ax) = LX=
a* =
k,
iff
ml = 1,
that (7) is the
A. Theorem
2 for
A was proven already in [4].
of the spectral
radius of positive
operators
cones, and [3] for related results.
REFERENCES A. Berman
and R. J. Plemmons,
Academic,
New York,.1979.
G. P. Barker and H. Schneider,
Appl. 11:219-233 K. H. Fijrster
S.
matrix,
Characterizations
Friedland,
theory,
numbers
Linear Algebra
and the local spectral
Linear Algebra Appl. 120:193-205 of the spectral
radius of positive
(1989). operators,
Linear
operators
on
(1990).
Characterizations
of spectral
radius
of positive
C*
/. Funct. Anal., to appear.
S. Karlin, Positive
operators,
J.
M. G. Krein and M. A. Rutman,
(1959).
Math. Mech. 8:907-937 Linear operators
space, Uspekhi Mat. Nauk 3:95 (1948); G. K. Pedersen,
leaving invariant cone in a Banach
Amer. Math. Sot. Transl.
No. 26.
C*-Algebras and Their Automorphism Groups, Academic,
B. S. Tam and S. F. Wu, preserving
Perron-Frobenius
and B. Nagy, On the Collatz-Wielandt
Algebra Appl. 134:93-105 algebras,
Algebraic
(1975).
radius of a nonnegative S. Friedland,
Nonnegative Matrices in the Mathemutical Sciences,
On the Collatz-Wielandt
map, Linear Algebra Appl. 125:77-95
sets associated
(1989).
1979.
with a cone
754
FRANK UHLIG,
TIN-YAU TAM, AND DAVID
EXTREME DOUBLY NONNEGATIVE
CARLSON
MATRICES WITH PRESCRIBED
ROW SUMS by ROBERT GRONE35
Let either
Zf, denote
the convex
the real or the complex
extreme
cone
of all n-by-n
positive
semidefinite
case, it is obvious that a matrix
ray in H, if and only if the rank of A is 1. Several
structure
of the set of extreme
example,
the
diagonal. points,
correlation
The extreme have been
set the notation,
for example,
rays of subcones that
familiar convex
convex
(undirected,
interest
set of matrices
set in
obtained
by establishing
is of some
H,
matrices
matrices,
the ranks of between
the
this result being
and methods
interest.
if p = (p,,
employed K,
role.
previously
One useful
Our
in describing for simplicity,
mentioned
concept
a lot of
that it held for A in K,, techniques
extreme
be the set
. , p,. Although points of K,
and because
of extreme graph
are
some of the
points will be similar
in that rank and iero patterns
will be the
so
are valid in
> 0, let K(p)
. . . , pJT
papers deal only with K,. The investigation
the extreme
stimulated
in H, which have row sums pr,
we will use the notation
to that of the other problems an important
To
is a,,, the set of n-by-n doubly stochastic
conjecture
In particular,
nonnegative
valid for K(p),
authors
with no loops or
and there is a distinction
points of 62, are just the permutation
more generality.
of all entrywise
referenced
various
a given sparsity pattern.
some 40-odd years ago. In this note we wish to consider
this particular
For
on the
. . , n}. and let M(G) consist of all A in H, such that
activity on the van der Waerden
the theorems
l’s
ranks of extreme
[7], and [lo]
points of the set K, = a2, fl H,. In 1962, Marcus and Newman [ll]
somewhat
of H,.
have
cases.
The extreme
due to Birkhoff
subsets
which
i # j and (i, j) is not an edge in E. In these investigations
real and complex Another
authors have studied the
the possible
in [9]. In [l],
of H, which respect
points or rays are of particular
matrices.
In an
convex
in H,
C = (V, E) is a simple
multiple edges) graph on V = (1, extreme
matrices
points of this set, and in particular
suppose
aij = 0 whenever
or rays of certain
are those
matrices
studied,
studied the extreme
points
matrices.
A in H, generates
of a symmetric
play
matrix.
If
A = AT is n-by-n, we let G(A) be the graph on V = { 1,. . , n} with edges (i, j) for all i, j for which V = (1,.
i # j and aij # 0. If G, = (V, E,)
. , n}, we say that G, is a s&graph
we use the notation
and Fisher
in K,.
They established
This work
DMS-9007048.
matrix in K,
in H,. Furthermore,
35Department 92182.
noted that I, is always an extreme
in Q,, and that I,, is the only extreme
noted that the unique rank-l it is extreme
In this case
G, C G,.
In [5], Christensen it is extreme
and G = (V, Ea) are two graphs on
of G, if and only if El C E,.
(l/n)J,
has been
Sciences,
supported
since
They also
is (l/n)J,, and that it is extreme in K, since is the only entrywise positive extreme point
also that rank-2 matrices
of Mathematical
point in K,
point of rank n in K,.
are extreme
San Diego
by the
National
State
in K,
if and only if they
University,
Science
San Diego,
Foundation
under
CA grant
AUBURN
1990 CONFERENCE
have zero entries.
These
the authors
obtained
established
an inequality
ON MATRIX
observations
tell the complete
some tridiagonal
extreme
which relates
755
THEORY story for n = 2,3.
For n = 4,
points of rank 3. For general
the number
of nonzero
entries
n, they
of an extreme
point with its rank. These
results
determine
suggest
extremality
the possibility
for A in K,.
the rank of G be the minimum
that rank A and G(A)
If G is a connected
rank of A in K,
might suggest that for a given G, either no matrices or else the extreme
matrices
in K,
with G(A)
might be sufficient
graph on V = (1,.
with G(A) in K,
= G. The results
with G(A)
= G are exactly
to
. . , n}, let so far
= G are extreme,
those with rank A =
rank G. We shall see later that this is not the case. In testing any A E K, matrix C a perturbation
for extremahty,
the following notion from [8] is useful.
Call a
of A if and only if
(i) C = CT, (ii) Ce = 0,
A C nullspace C, and
(iii) nullspace
(iv) G(C) C G(A). These four constraints perturbation K,
space is trivial.
is replaced The
are linear and homogeneous,
of the n-by-n real matrices.
a subspace
A matrix
results
rank A = n criteria
of A form
The reader should note that this definition
is unchanged
in [5] give a characterization 1. We assume without
that
C(A)
of the extreme the structure
loss of much generality
is connected.
By counting
points
of extreme that
equations
A in K, A in K,
rank A = n -
with
and unknowns
or
in the
to exist, we saw that if rank A = n - 1 and A is extreme,
for a perturbation We established
with
A is irreducible,
then G(A) has less than n edges. This forces G(A) to be a tree, or equivalently, acyclic.
if
by K(p).
rank A = 1, 2, or n. In [8], we considered equivalently,
and so the perturbations
A in K, is extreme if and only if this
that when
G(A)
is a tree,
then
A is extreme
1. Also, if rank A = n - 1 and A is irreducible,
A to be
if and only if in K,
then A is extreme
if and only if G(A) is a tree. In [2] a lemma
was obtained
which
was also useful in [7j and [8]. The matrices
studied in [2] were doubly nonnegatiue, that is, positive semidefinite nonnegative. irreducible enabled
Clearly,
in K,
the matrices
and doubly nonnegative,
and G(A)
us in [8] to give a complete
bipartite.
and
K(p)
is bipartite,
answer to when
if A in K, has G(A) bipartite
Specifically,
as well as entrywise
fall into this category.
If
A is
then rank A 2 n - 1. This
A in K,
is extreme
and connected,
if G(A)
is
then A is extreme
ifandonlyinG(A)isatreeandrankA=n-1. In [8] we also considered exactly
the case when G is unicyclic
n edges and one cycle).
bipartite
and there
is no extreme
If the unique
(that is, G is connected
cycle in G is of even length,
with
then G is
A in K, with G(A) = G. If G has a cycle of odd
length, then any A in K, with G(A) = G has rank at least n - 2. In this case, then A in K, with G(A) = G is extreme if and only if rank A = rank G = n - 2. We also established
the existence
rank, n - 2. Showing result
for K(p)
of A in K,
which has this graph and also has the minimal
that this minimum
by invoking
rank is obtained
the DAD theorem
[4]. Lastly,
in K,
also establishes
the
we noted that if G is any
756
FRANK UHLIG,
graph on V = (1,.
TIN-YAU TAM, AND DAVID
. . , n} for which there is a matrix
A in K,
rank A = k, then for every m, k < m 4 n there exists a matrix andrankB=m. In [6], we considered Say that agraph common
edge.
a class of graphs which generalizes
G on V= Suppose
{l,...,
n} is nonchordal
G is a connected
cycles
and
nonchordal
has exactly
r even
s odd cycles.
corresponds
to a tree, and that r + s = 1 corresponds
results in [8] can be used to establish
an induction
that rank G = n - s - 1, and we constructed minimal
rank. Again, the DAD theorem
has G(A) nonchordal
connected
such a matrix is extreme
Berman
graph on V = (1,.
. , n} which
that the case
in K,
example
points
points in K, lays to rest
qualitatively Lastly,
when
r = s = 0 Hence
with specified
this to K(p).
Suppose
and s odd cycles.
that rank A and G(A)
for n = 4. For completely
n = 5 this pattern
determine
different
possibility
of a nice,
points
in K,.
for certain
p.
the presentation
in [l], [7j, and [lo]. of these
exactly the same if K,
results
characteriIn [3],
of two matrices
simple
answer
investigations
to the K,
problem
and
are closely
the extremality
in K,
and the other not. This
As R. Brualdi quite correctly
in Auburn,
is replaced
1.
to determine
fails to hold.
It may even turn out that
we would like to note that these
sparsity questions
A in K,
We found that
the rank and sparsity possibilities
for n = 5. They also give an example
the
extreme
the
graph and
that
are sufficient
with equal ranks and the same graph, but where one is extreme characterizing
graphs.
of G share a
on r + s. For such a graph we found
with r even cycles
suggests
and Shaked-Monderer
of extreme
trees and unicyclic
= G
if no two cycles
This is true for n < 4, and our results in [6] provide a complete
zation of the extreme
= G and
to G being unicyclic.
matrices
extends
with G(A)
B in K, with G(B)
if and only if r = 0 and rank A = rank G = n - s -
So far the evidence extremahty.
Note
CARLSON
related
pointed
question
K(p)
of are
to the
out during
for A in K,
is
with K, Il M(G( A)).
REFERENCES J. Agler, J. W. Helton,
S. McCullough,
and L. Rodman,
Positive
definite
with a given sparsity pattern, Linear Algebra Appl. 107:101-149 A. Berman and R. Crone, Bipartite completely positive matrices, Philos. Sot. 103:269-276 A. Berman
(1988).
and N. Shaked-Monderer,
More
doubly stochastic matrices, preprint. R. Bruahh, The DAD theorem for arbitrary 45:189-194 J.
points,
R. Grone, matrices, R.
Grone
on extremal row sums,
positive
Proc.
semidefinite
Amer.
Math. Sot.
(1974).
Christensen
extreme
matrices
(1988). Proc. Cambridge
and P. Fisher,
R. Loewy,
and S. Pierce,
Linear and Multilinear and
Positive
Linear Algebra A&.
S.
Pierce,
131:39-50 (1990). R. Grone and S. Pierce,
Extremal
doubly (1986).
Nonchordal
positive
stochastic
matrices
semidefinite
and
stochastic
Algebra, to appear.
Extremal
Linear Algebra Appl. 150:107-117
definite
82:123-132
bipartite
positive (1991).
matrices,
semidefinite
Linear
Algebra
doubly stochastic
Appl.
matrices,
AUBURN 9
1990 CONFERENCE
R. Crone,
ON MATRIX
757
THEORY
S. Pierce,
and W. Watkins, Extremal correlation matrices, Linear (1990). J. W. Helton, S. Pierce, and L. Rodman, The ranks of extremal positive semidefinite matrices with given sparsity pattern, SZAMJ. MatrZx Anal. 10:407-423 (1989). M. Marcus and M. Newman, Inequalities for the permanent function, Ann. of Math. 75:47-62 (1962). B. Ycart, Extremales du cone des matrices de type non negatif a coefficients Algebra Appl. 134:63-70
10 11 12
positifs ou nuls, Linear Algebra AppZ. 48:317-330
RECENT
RESULTS
by SHU-AN
ON THE
(1982).
PERMANENTAL
HU36 and TIN-YAU
NUMERICAL
RANGE
TAM37
The purpose of this synopsis is to give an account of recent advances on the permanental numerical range. The kth permanental numerical range of A E enxn (the set of n x n complex matrices),
where 1 < k < n, is defined as
Pk( A) = {per(U*AU)IUe$,,k,
U*U = Zk},
and per B = CoeS, Htr bj,(i) is the permanent function for BE @kxk. This definition is motivated by the classical numerical range of A E B,xn:
W(A)
If k = 1, then Pr( A) = W(A). upon five properties
= {x*Axlz~V,
IIxII = 1).
There are many nice results for W(A).
here and investigate generalizations
We shall touch
to the permanental
numerical
range. Maybe the most interesting W(A)
one is the. celebrated
Toeplitz-Hausdortf
theorem:
i.sconoexfwanyA~@&,.“.
In particular,
there is a complete
Let A E @2zx2 have eigenvahm
description
Xl and &.
for the shape of W(A)
Then W(A)
when n = 2:
is an elliptical disk with foci
at A, and &,, minor axis of length
dtr(A*A) -
l&l’-
l&l’,
3sDepartment of Mathematics, University of Connecticut, Storrs, CT 06269. 37Department of Algebra, Combinatorics 5307.
and Analysis, Auburn University, AL 36849-
758
FRANK
UHLIG,
TIN-YAU TAM, AND DAVID
CARLSON
and major axis of length
Vtr( In particular,
A*A)
- 2 Re( A,&)
if
c
Al o
A=
hz>
[
1
A, and &,
Recently,
the authors
[5] obtained
Let ~~~~~~
THEOREM 1.
the following
have eigenvalues
minor axis of length
1c 1, and
analogy for Ps( A):
A, and AZ. Then Pz( A) is an elliptical
disk with foci at X,X, and i( XT + hi), minor axis of length
d[tr(
A*A)
-
] X, ] ’ -
] X, ] “] [tr( A*A)
- 2 Re( A1x2)]
,
and major axis of length
tr( A*A)
In particular,
-
IhI2
2
-
-
l&l2
if
Al
A=
0
c
then P2( A) is an elliptical disk with foci at AlA2 IA,-X,12+
= (0)
In [8] Marcus
ifand
W(A)
Recently
Jc12+
)A,-A212/2.
enjoys is:
only ifA
= 0.
and Wang asked whether
they posted it as a conjecture.
THEOREM 2.
and +(A; + A$, minor axis of length
Ic(2,andmajoralrisofZength
Another property W(A)
1
%>
[
ICI
- 2Re(Xr&).
2
Let A E G,,,,
Chan [l] extended
Then
the above property
is true for Pk( A), and
Hu [4] proved the conjecture
1 < k Q n. Then Pk( A) = (0)
the result to arbitrary
ter. Let 1 ,< k < n, and let G < Sk be a subgroup
subgroups
ifand
in the affirmative: only ifA
with principal
of the full symmetric
= 0. charac-
group Sk of
AUBURN degree
1990 CONFERENCE
k, and x
759
ON MATRIX THEORY
: G -+ G an irreducible
character
of C. Then define the following
range:
P,“(A)
= {~;(U*AU)IUEC”,~,
where d:(B) = C oeC x(u)H!=, bio(i), associated with x. Chan [l] obtained:
for
BE
U*U = Zk},
ejkxk~
is the generalized
matrix function
THEOREMS (a) Zfx 3 1, then P:(A)
= {0}
ifand
only ifA = 0.
(b) Zf x f 1, then rank A < 1 implies P,“( A) = (0). The third property
we want to discuss is:
W( A) is a line segment if and only if 5 A is Hermitian for some .$ E G with
1[ 1 = 1.
In [5] the authors obtained a similar result for Pk( A):
LetAEGnx,,.
THEOREMS.
(a) Zf n = k = 2, then Pz( A) is a line segment if and only if A is normal. (b) If 1 < k < n, excepting n = k = 2, then the following statements are equivalent: (i) Pk( A) is a line segment. (ii) Pk( A) is a line segment on a ray containing (iii)t A is Hermitian for some nonzero
COROLLARY
the origin.
t E G with
1.$1 = 1.
&AE@$,,,.
1.
(a) Zf n = k = 2, then P2( A) lies on the real axis (nonnegative real axis; positive real axis, respectively)
if and only if
(i) A or iA is Hermitian ;)
(A or -A
is positive semidefinite;
positive definite, respectively),
A is uniturily similar to diag(a + bi, a - bi) or diag(a + bi, - a + bi), where a and
b are
nonzero
real
numbers
(A is unitarily
similar
to diag(a + bi, a - bi),
where
a2 - b2 > 0; a2 - b2 > 0, respectively). (b) Zf 1 < k Q n, excepting n = k = 2, then Pk( A) lies on the real axis (nonnegative real axis; positive real axis) if and only if 5 A is Hermitian for sollze 5 E G such that tk = 1 or - 1 (.$ A is positioe semidefinite;
positive definite, fm some k th root of unity
.$, respectively). The fourth property
is:
Zf A is positive semidefinite, largest eigenvalues
as endpoints.
then W(A)
is a line segment with the smallest and the
FRANK UHLIG,
760 It is a long-standing semidefinite,
of U*AU are equal.
[lo]
to find max{
min{ 1z 1 I .z EP,( A)}
whereas
Mehta [9] conjectured conjecture,
problem
TIN-YAU TAM, AND DAVID
that the maximum Recently
CARLSON
1z 1 ) z E P,( A)} if A is positive
= det A is well
known
(see
e.g.
is attained when all the main-diagonal
Drew and Johnson
[3] gave a counterexample
[lo]). entries
to Mehta’s
and in [2] they proved that:
THEOREM 5. If A is a 3 x 3 real positioe semideftnite matrix, then max{ I .zI I z E P3( A)} is always of the form $A,( A! + A:) or $[3Ai(h, + A,) + A\ + A” + (% + A\ Ai)3/2] for sine ordering of the eigenvalues.
It turns out that there exists a persymmetric the maximum persists
(i.e.,
symmetric
with respect
to the
diagonal as well as the main diagonal) matrix U*AU that yields
upper-right-to-lower-left
for n = 2,3.
It was then asked in [3] whether
the persymmetry
criterion
for n 2 4.
Lastly, in [ll]
Pellegrini proved that:
A linear operator T: G,x, -+ Ctnxn satisfws W(T( A)) = W(A) and only if there exists U E U,(G) such that
for all A E enxn
if
(i) T(A) = U*AU for all A E Gnxn or (ii) T(A) = U*ATU for all A E GnX,. In [5, 121 the authors extended
the result.
Let T: Gnxn + Gnxn be an operator. THEOREM 6. Pk( A) for all A E C?,x, if and only if: Case I.
Then T satisfis
If 1 6 k < n, excepting k = n = 2, there exist U E U,(c)
Pk(T( A)) =
and a k th root of
unity 5 such that either (i) T(A) = f;U*AU for all A E Gnxn, or (ii) T(A)
= tU*ATUfw
If k = n = 2, there exists UE U,(G) such that either
Case 2. (i) T(A)
all AE@,~,,.
= ~U*AU~~~U~EAE@&~,,
(ii) T(A) = +U*ATUfor (iii) T(A)=
+-I ‘iA
(iv) T(A)=
+ y’l+““*(
REMARK.
2 +iU*
Recently,
or
all AE@&~~, ( AATLei
or
CA 2
z)
yIS)
[7] obtained
UforallAEGZxZ,
or
UforaZlAEG2x2. part
of Corollary
1 independently.
recently the authors [6] extended Theorem 4 and Corollary 1 for arbitrary Sk with principal
More
subgroups
of
character.
REFERENCES 1
C.
T.
Chan,
M&linear
Some
more
on a conjecture
Algebra 25:101-105
(1989).
of Marcus
and Wang,
Linear
and
AUBURN 2
1990 CONFERENCE
J. H. Drew and C. R. Johnson,
The maximum
semi-definite matrix, given
eigenvalues,
25:243-251 3
J.
H.
4
the
and C.
R. Johnson,
permanental
(1989). S. A. Hu,
On
20:191-196
(1987).
the
Counterexample
and
of a 3-by-3
positive
M&&near
Algebra
Marcus-Wang
conjecture,
S. A. Hu and T. Y. Tam, Operators
6
line, to appear. S. A. Hu and T. Y. Tam,
to a conjecture
Linear and Mu&linear
maximization,
5
character,
permanent Linear
(1989).
Drew
regarding
761
ON MATRIX THEORY
Linear
with permanent
On the generalized
of Mehta
Algebra 25:253-254
and M&linear
numerical
numerical
Algebra
ranges on straight
range with principal
preprint.
Tian-Gang Lei, On the numerical M. Marcus and B. Y. Wang, Mu&linear Algebra 9:111M. L. Mehta,
range of an induced power, preprint.
Some variations
on the numerical
range,
Liner and
120 (1980). Hindustan
Elements of Matrix Theuy,
Publishing,
Delhi, 1977,
p.
156. 10
H.
Mint,
21:109-148 11
Theory
of permanents,
V. J. Pellegrini, Numerical T. Y. Tam,
COMPLETELY
KOGAN3’
applications
range of induced
MATRICES
Algebra
n x m matrix.
positive matrices
include
on a Banach algebra, St&a
power,
Linear and Multilinear
AND GRAPHS
and ABRAHAM
is called the factorization
Completely
operators
BERMAN3”
3g
A is completely positive if it can be decomposed
B is a nonnegative
factorization
and M&linear
(1988).
POSITIVE
An n x n matrix where
range preserving
On the numerical
Algebra 23:207-211
by NATALIA
Linear
(1975).
Math. 54:143-147 12
1982-1985,
(1987).
“a proposed
sectors of the U.S. economy”
The minimal number
as A = BB’,
m that admits such a
index of A and is denoted by q(A).
are important in the study of block designs [q. Other mathematical
model of energy demand
for certain
and statistics [5].
A matrix which is both elementwise
nonnegative
and positive semidefinite is called
doubly nonnegative. It is obvious that every completely
positive matrix is doubly nonnegative,
but the
converse is not always true [3, 5, 61. It depends in some sense on the zero pattern of the matrix. To describe this dependence,
we associate with an n x n symmetric
38Department of Mathematics, Technion-Israel Israel.
matrix
A a
Institute of Technology, Haifa 32000,
3QResearch supported by the C. Wellner Research Fund at the Technion.
762
FRANK
graph G defined by V(G) graph G is completely completely positive.
= { 1,
positive
UHLIG,
TIN-YAU TAM, AND DAVID
CARLSON
,n}, E(G) = {(i, j) : i # j, aij # 0). We say that a
if every doubly nonnegative
The aim of this work is to characterize
completely
matrix
positive
A with G(A) graphs.
= G is
The following
results are known:
PROPOSITION1 [5, 91.
A graph
PROPOSITION2 [2]. completely
Bipartite
graphs
(graphs
positive i$n < 5.
is completely
which
contain
no odd
cycle)
are
positive.
PROPOSITION3 [3]. If a graph then G is not completely The characterization A is completely
m-dimensional
vectors,
doubly nonnegative tive inner products, that the coordinate
G contains
an odd cycle
of length
greater
4,
than
positive. is completed
when G is not completely matrix
G with n vertices
positive. positive where
by proving
that Proposition
that an n x n
if and only if it is the Gram matrix of n nonnegative m may be greater
than
n. Using the fact that every
matrix is a Gram matrix of a set of vectors we find an m-dimensional vectors
3 is the only case
The proof is based on the observation
with mutually nonnega-
space with an orthonormal
basis such
of the set in this basis are nonnegative.
To prove our main result we use the following lemmas:
LEMMA 1. Let k be a cutpoint {k}.
If both
G, and G,
LEMMA 2.
of a graph
are completely
The graph
G, i.e.,
positive,
T, consisting
G = G, U G,
and G, fl G, =
then so is G.
of n triangles
with a common
base is completely
positive. Based on the above facts, we obtain
THEOREM 1.
Let G be a nondirected
graph
without
equivalent : (1) G is compl&ely
positive.
(2) G has no odd cycle of length greater (3) G consists
of blocks of the following
than 4. types:
(a) blocks with less than 5 vertices; (b) bipartite (c) families
blocks; of triangles
with the common
(4) All the blocks of G are completely (5) G is the root graph of a pafect
base.
positive.
line graph.
loops.
Then the following
are
AUBURN
1990 CONFERENCE
The proofs of the lemmas and of the equivalence A slightly different equivalence
proof
of the equivalence
of(l),
(2), and (3) are given in [S].
of (2) and (3) is given
in [4]. The
of (2) and (5) is given in [lo].
Let G be a nondirected completely
763
ON MATRIX THEORY
graph without loops. The set of all factorization
positive realizations
indices of
of G is denoted by Z(G). In the next theorem
we list
some facts we know about Z(G).
THEOREM 2. U G, such that G,, . . . , G, are connected by a cutpoint (see Lemma l), then Z(G) = Z(G,) + *.. +Z(G,) + {O,l}. (2) Zf K, is a complete graph, G(A) = K,, and A is a completely positive matrix, then (p(A) = rank A, and thus Z(K,) = (1, . . , n}. (3) If ) V(G)) = n Q 4 then Z(G) E {l,. . . , n}. (4) IF G is bipartite then Z(G) E { ( E(G) (; ( E(G) ( + 1). Zf G is a tree then (1) Zf G = G, U ...
Z(G) = { I E(G) I + 1). (5) ZfG=T,(seeLemma2)thenZ(G)S{n,...,2n+l}. (6) Zf there are k independent vertices in G, then Z(G) fI (1,.
. . , k} # 0.
Results (I), (2), and (5) could be improved by settling the following question:
QUESTION. Let a, b E Z(G). Does
Z(G) contain all the integers between
a and b?
The authors would like to thank Professor T. Ando for suggesting matrix theoretic proofs, using Schur complements, of Lamas Z and 2 [Z].
REFERENCES 1
T. Ando, private communication.
2
A. Berman and R. Grone, Bipartite completely
PhiZos. Sot. 103:269-276 3
A. Berman matrices,
4
at
Jerusalem,
Maximum
the
Combinatorial
k-coloring
French-Israeli
Proc. Cambridge
results
on completely
positive
(1987).
of perfect
line graphs and their roots, pre-
Symposium
on
Combinatorics
Nonnegative
factorization
and
Algorithms,
Nov. 1988.
L. J. Gray and D. G. Wilson, nonnegative
6
Hershkowitz,
Linear Algebra Appl. 95:111-125
Gavril Fanica, sented
5
and D.
positive matrices,
(1988).
matrices,
Linear Algebra AppZ. 31:119-127
M. Hall, Jr., A survey of combinatorial
of positive semidefinite (1980).
analysis, in Surueys Appl. Math. IV, Wiley,
1958, pp. 35-104. 7
M. Hall, Jr., Cumbinatorid
8
N. Kogan and A. Berman, Math., to appear.
Theory, Blaisdell, Lexington, Mass., 1967. Characterization
of completely
positive graphs, Discrete
764 9
FRANK UHLIG,
TIN-YAU TAM, AND DAVID CARLSON
J. E. Maxtleld and H. Mint, On the equation Sot. 13:125-129
10
L. E. Trotter,
RANK
Line perfect graphs,
PRESERVERS
by RAPHAEL
X’X
= A, Proc.
Edinburgh
Math.
(1962).
AND
Math. Programming
INERTIA
12:255-259
(1977).
PRESERVERS
LOEWY4’
Let V be a vector space which is one of the following: (1)
P”,
the
throughout
set of all m x n matrices
with entries
in a field
F.
We assume
that m Q n and F is infinite.
(2) H,, the set of all n x n Hermitian
matrices.
(3) S,, the set of all n x n real symmetric Given a matrix
matrices.
A, let p(A) denote the rank of A. Let
ttj=
{AEV:~(A)
For A E H,, or A E S,, let In A = (r, s,
=j).
t), where r is the number of positive eigenvalues t the number of zero eigenvalues
of A, s the number of negative eigenvalues of A, and of A. Let
C( r, s, t) = { A : In A = (r, s,
We
assume
throughout
that
k is a fixed positive
t)} .
integer
and T
: V + V a linear
transformation.
DEFINITION
(a) We say T is a rank-k preserver
if p(A) = k implies that p(T( A)) = k, i.e., if R,
is invariant under T. (b) We say T is rank-k nonincreasing
if p(A) = k implies p(T( A)) Q k. It is easy to
see that (under our assumptions) this is equivalent to the statement that the set lJf=,
Rj
is invariant under T. (c) If V = H,, or S,, we say T is a G(r, s, t)-preserver
if the set G(r, s,
t) is
invariant under T. We consider here the following three problems, which seem to have attracted of interest in recent years.
40Department of Mathemaics, Technion, Haifa 3200, Israel.
a lot
AUBURN
1990 CONFERENCE
ON MATRIX
PROBLEM
1.
When is T a rank-k nonincreasing
PROBLEM
2.
When
PROBLEM
3.
When is T a G(r, s, t)-preserver?
In thefrrst
765
THEORY map?
is T a rank-k preserver?
two problems
we assume V = Fmv”, while in
the third we assume V = H,, or
SIL.
Problem 1 A full solution functional
is known
THEOREM
1[12].
PI)
(III)
T(A)
= L’AtV
T(A)
=
(IV) It should be noted course,
nonincreasing
T is rank-l
I’( A) = UAV
(1)
depend
only for the case
k = 1. We denote
by (p(A) a linear
on A. Botta showed:
on
forsome
[a,
that in (III)
A. As indicated,
some related
THEOREM
U, VEF”‘,“,
ez(A)
a2
...
...
dA$
a,].
and (IV) ai are fixed elements
no analogous
it is clear that if T satisfies
VEF”,“,
UEF”‘,“‘,
for some
[v&4)
q and only if it is one of the following:
statement
of F and do not
holds for any arbitrary
(I) or (II), then it is rank-k nonincreasing.
k. Of
We state
results. 2 [lo]. Suppose F is algebraically
closed and T is rank-k nonincreasing.
Then either Im T c lJj”=, Rj or dim Ker T < mn - (k + 1)2. THEOREM 3 [8, 131. Suppose F is algebraically closed. If k < m and T(Ujk_ 1 Rj) c UT=, Rj, then either (I) holds, or m = n and (II) holds, where U and V are nonsingular. Note that the assumption a significant
restriction
on T in Theorem
3 means that Ker T n { UJ=, Rj} = 0,
on the rank-k nonincreasing
map T.
766
FRANK
UHLIG,
TIN-YAU TAM, AND DAVID
The following result is useful in the investigation was also used by Loewy in the investigation
THEOREM 4 [18].
CARLSON
of rank-k nonincreasing
of Problem
maps, and
3.
Suppose that T is rank-k nonincreasing.
Then it is rank-l nonin-
creasing for every 1 > k. Using this theorem,
Loewy has recently
THEOREM 5 [22]. nonincreasing,
proved the following:
Suppose that F is algebraically
closed and k < m. If T is rank-k
and Im T contains a matrix B such that p(B) 2 k + I, then either (I) or
(II) must hold. It is now clear, in light of Theorem set of rank-k nonincreasing
said if T is rank-k nonincreasing
DEFINITION.
5, that in order to completely
A subspace
It is clear that given any l-subspace For
example,
can
one
if p(A) < k for all A EL.
L, one can build rank-k nonincreasing
it is desirable
characterize
the
What can be
Q k for all A E lm T? This leads us to:
and p(A)
I_. is said to be a x-subspace
whose image is L. Therefore inclusion)?
characterize
maps we are faced with the following question:
the
to obtain information
maximal
i-subspaces
about (with
maps
Z-subspaces.
respect
to set
The task seems quite formidable.
Atkinson
and Lloyd [2, 31 and Atkinson
They defined
the concepts
weak canonical Eisenbud
of primitive
form for a %-subspace.
and Harris
roughly equivalent
[14]. They
[l] obtained
and imprimitive Another
recent
on l-subspaces.
Z-subspaces,
and obtained
paper on z-subspaces
state that the problem
to the problem
some results
of classifying torsion-free
%subspaces
of classifying
certain
the maximal
dimension
of a &subspace
of F”‘,“.
Then dim L Q kn.
a
is due to is
sheaves on projec-
tive spaces. The
problem
Flanders
showed
of determining
THEOREM 6 [15]. Flanders
Suppose L is a x-subspace
also characterized
the case where
equality
is attained.
is easier.
He assumed
that
1F 1 > k + 1, and for the case of equality also char F # 2. Meshulam [25] reproduced Flanders’s
results,
removing
the restrictions
on F. His proof uses the KGnig-Egervary
theorem. PROBLEM 2.
We assume that F is algebraically
holds, or m = n and (II) holds, where preserver. direction
The question was obtained
THEOREM 7 [24].
here is whether
closed.
It is easy to see that if (I)
U and V are nonsingular, the converse
is true.
then
T is a rank-k
The first result
in this
by Marcus and Moyls.
Suppose that T is a rank-l
preserver.
m = n and (II) holds, where U and V are nonsingular.
Then either
(I) holds, or
AUBURN
1990 CONFERENCE
Marcus
ON MATRIX
and Moyls assumed
767
THEORY
that char F = 0. Westwick
[29] obtained
the same
result for char F # 0. In their paper, Marcus and Moyls also raised the following: CONJECTURE1.
The conclusion
of Theorem
7 holds for a rank-k preserver,
where
k is any integer such that 0 < k < m. This conjecture is to date not completely resolved. We give some partial results, all confirming the conjecture. Moore [26] proved the case k = 2. Beasley [5, 7, 91 obtained various results.
They include
the confirmation
of Conjecture
1 in the cases
k = 3,
k = m, and k < $n. Recently Beasley has been able to show: THEOREM8 [ll].
Conjecture 1 h&s
solution in case F = a? It turns out that in the problem
Why do we get a complete of characterizing
rank-k preservers,
role, much as z-subspaces
a certain
family of subspaces
are associated with rank-k nonincreasing
plays an important maps.
L of Fmq” is said to be a k-subspace if p(A) = k for any
DEFINITION. A subspace
AEL,
if F = c.
AfO.
There are several papers dealing with k-subspaces.
Earlier
ones are due to West-
wick [3O] and Beasley [6]. They obtained bounds for the dimension of these subspaces. Beasley, for example, showed that if L is a k-subspace then dimL
F = @ is made. His result is somewhat too invol&l
based on Sylvester’s
work, Westwick
to be stated here.
[31] was able to show that if L is a
k-subspace of G”‘* ” then dimLgm+n-2k+l. Based on this inequality, Beasley was able to cover (for F = G) the cases k > fn that were not covered
PROBLEM 3.
in his earlier work, thus obtaining Theorem Suppose that (r, s,
t) is a fixed inertia triple, and let T : H,, + H,.
Suppose that there exists a nonsingular
(“1
8.
S E en*” and E such that either
T(A)
= ES*AS
Or
(“I)
T(A)
= ES*A’S,
where E = 1 if r # s and E = f 1 if r = s. Then T is a G(r, s, t)-preserver.
768
FRANK UHLIG,
TIN-YAU TAM, AND DAVID CARLSON
The obvious analogue holds for S,. Johnson and Pierce showed:
THEOREM 9 [17].
Suppose
that T
: H,, --) H,, is an invertibb G( r, s, t)-preserver. 0, 0), (0, n, 0), (0, 0, n), (n /2, n/2,0). Then either
Suppose that (r, s, t) is not one of (n, (V) or (VI) must hold. The corresponding
result for S, also holds. What about the four exceptional
The set G(O,O, n) consists of 0, so it is of no interest.
classes?
We clearly have G(0, n, 0) =
- G( n, 0,O). The class G( n, 0,O) consists of all n X n positive definite matrices, set of G(n, O,O)-preservers definite matrices
consists
so the
of all linear maps that map the set of positive
into itself. This is a well-known
open problem.
Pierce
and Rodman
showed:
THEOREM 10 [27]. invertible G( n/2,
Suppose
n/2,0)-preserver.
n is an even integer,
n > 4, and T
Pierce and Rodman also characterized
the set of G(l, 1, 0)-preservers,
contains the set given by (V) and (VI). The proof of Theorem proof
of Theorem
subspaces
of en,
9.
It uses the Grassmannian,
and the gap metric
the real symmetric
THEOREM 11 [20].
Suppose
such that T(A)
elements
are the
It should be noted that
n > 4, and T : S, -+ S, is an
Then there exist nonsingular
11 uses a result of Friedland
- I + 2) contains a nonzero
S E R n, n and E = f 1
9, 10, and 11 assume
Loewy and Pierce
1, any subspace
and Loewy [16] which states L of S, with dim L > i(l -
matrix whose largest eigenvalue has multiplicity
least 1. The method of proof of Theorem Theorems
whose
= ES~AS.
The proof of Theorem
dropped?
which strictly
10 is different from the
10. However, we managed to prove
n is an even integer,
that given any 1 such that 2 Q I Q n I)(2n
a space
is put into this space.
case is not covered by Theorem
invertible G( n 12, n /2,0)-preserver.
: H,, + H,, is an
Then (V) or (VI) must hoM.
11 can be used to prove Theorem that
T is invertible.
at
10 as well.
Can this assumption
be
[23] gave a positive answer for the class G(l, 1, n - 2) in
case n 2 3. Johnson and Pierce [17] gave a positive answer for the classes G(n and G(k + 1, k,O), and therefore
also for G(l, n - 1,0)
1, 1,O)
and G(k, k + 1,O). They also
stated:
CONJECTURE 2.
Suppose that n > 3 and rs > 0. If T is a G(r, s, t)-preserver,
then
either (V) or (VI) must hold. We have
THEOREM 12 [21].
Conjecture
The proof of Theorem earlier. The assumption class
G(r, 0, n - r),
2 holds ifr
12 relies heavily on rank-k nonincreasing
rs > 0 in Conjecture
where
# s.
0 < r < n. The
2 is essential. map
T
maps, discussed
Indeed, consider now the
: H, + H, defined by T(H) =
AUBURN 1990 CONFERENCE
ON MATRIX THEORY
(tr H)I,. $ 0 is easily seen to be a singular G( r, 0, n - r)-preserver. showed:
THEOREM13 [4]. Let T : H,, + H,, be a G( r, 0, n - r)-preserver, p(T) > r2. Suppose that 0 < r < n. Then (V) or (VI) must hdd.
769 Baruch and Loewy
and suppose that
It can be shown that the bound r2 cannot be improved. It comes from the possible dimensions of faces of the cone of n x n positive semidefinite matrices in H,, which are 12, 1 = O,l,. . . n. In the real symmetric case r2 should be replaced by $-(r + 1). The case r = 1 was proved earlier by Loewy [19].
REFERENCES 1
2 3 4 5 6
M. D. Atkinson, Primitive spaces of matrices of bounded rank II, J. Au&d. Math. Sot. Ser. A 34:306-315 (1983). M. D. Atkinson and S. Lloyd, Large spaces of matrices of bounded rank, Quart. J. Math. Oxford 31:253-262 (1980). M. D. Atkinson and S. Lloyd, Primitive spaces of matrices of bounded rank, J. Au&d. Math. Sot. Ser. A 30:473-482 (1981). M. Baruch and R. Loewy, in preparation. L. B. Beasley, Linear transformations on matrices: The invariance of rank k matrices, Linear Algebra Appl. 3~407-427 (1970). L. B. Beasley, Spaces of matrices of equal rank, Linear Algebra Appl. 38:227-237 (1981).
7
L. B. Beasley, Linear transformations which preserve fixed rank, Linear Algebra AppZ. 40:183-187 (1981). 8 L. B. Beasley, Linear transformations on matrices: The invariance of sets of ranks, Linear Algebra Appl. 48:25-35 (1982). 9 L. B. Beasley, Rank-k preservers and preservers of sets of ranks, Linear Algebra Appl. 55:11-17 (1983). 10 L. B. Beasley, Linear transformations preserving sets of ranks, Rocky Mountain /. Math. 13:299-307 (1983). 11 L. B. Beasley, Linear operations on matrices: The invariance of rank-k matrices, Linear Algebra Appl. 107:161-167 (1988). 12 P. Botta, Linear maps preserving, rank less than or equal to one, Linear and Mu&linear Algebra 20:197-201 (1987). 13 G. H. Chan and M. H. Lim, Linear transformations on tensor spaces, Linear and Multilinear Algebra 14:3-9 (1983). 14 D. Eisenbud and J. Harris, Vector spaces of matrices of low rank, Adu. in Math. 70:135-155 (1988). 15 H. Flanders, On spaces of linear transformation with bounded rank, J. I,.ono!on Math. Sot. 37:10-16 (1962). 16 S. Friedland and R. Loewy, Subspaces of symmetric matrices with a multiple first eigenvahie, Pa&c J. Math. 62:389-399 (1976).
770 17
FRANK UHLIG, C. R. Johnson and S. Pierce,
TIN-YAU TAM, AND DAVID CARLSON
Linear maps on Hermitian matrices: The stabilizer of
an inertia class II, Linear and M&&near 18
T. J. Laffey and R. Loewy, Linear and M&linear
19
Linear
Algebra 26:181-186
R. Loewy, Linear transformations Appl. 121:151-161
Algebra 19:21-31
transformations
(1986).
which do not increase
rank,
(1990).
which preserve or decrease rank, Linear Algebra
(1989).
20
R. Loewy, Linear maps which preserve a balanced nonsingular inertia class, Linear
21
R. Loewy,
Algebra Appl. 134:165-179 Linear
Appl. 11:107-112 22
(1990).
maps which preserve
an inertia class,
SIAM ].
Matrix Anal.
(1990).
R. Loewy, Linear mappings which are rank-k nonincreasing,
submitted for publica-
tion. 23
R. Loewy and S. Pierce,
24
M. Marcus and B. N. Moyls, Transformations Math. 9:1215-1221 (1959).
25
R. Mesbulam,
of matrices,
spaces,
Pacifi
Quart.
J.
J.
Math.
(1985).
C. F. Moore, Characterization a Tensor
on tensor product
On the maximal rank in a subspace
Oxford 36:215-229 26
unpublished.
Product
Space,
of Transformations
Univ. of British
Preserving
Columbia,
Rank Two Tensors of
Vancouver,
B.C.,
Canada,
1966. 27
S. Pierce and L. Rodman, Linear preserves balanced inertia, SZAM I.
28
J. Sylvester, conditions,
On the dimension
R. Westwick,
30
(1967). R. Westwick, 5:49-64
31
on tensor
satisfying rank
(1986). spaces,
Spaces of linear transformations
Pucifz
J.
Math.
23:613-620
of equal rank, Linear Algebra Appl.
(1972).
R. Westwick, 20:171-174
MATRICES INEQUALITIES by ROY
Transformations
(1988).
of spaces of linear transformations
Linear Algebra Appl. 78:1-10
29
of the class of Hermitian matrices with
Matrix Anal. Appl. 9:461-472
Spaces
of matrices
of fixed rank,
Linear
and M&linear
Algebra
(1987).
WITH
POSITIVE AND
DEFINITE
LINEAR
HERMITIAN
PART:
SYSTEMS
MATHIAS41
Let M,(C) real] matrices.
[respectively, We call A EM,
M,(R)]
denote the space of n x n complex [respectively,
positbe definite (respectively,
positioe semideftnite) if A
41Department of Mathematics, The College of William and Mary, Williamsburg, VA 23187. Research supported by an Eliezer Naddor postdoctoral fellowship in the Mathematical Sciences from the Johns Hopkins University during the year 1989-90 while the author was in residence at the Department of Computer Science at Cornell University.
AUBURN
1990
is Hermitian
CONFERENCE and
Hermitian
part
ON MATRIX
x*Ax > 0 (respectively,
and the skew-Hermitian
further
definite
results
definite
can be found
largest
Frobenius
We write
norm
use
that
A,,
being
have many
some
[respectively,
1
x
1
analogous
in this
synopsis.
and
to those Proofs
and
norm
the
alge-
. 11 2) and
(I]
semidefinite.
His proof
A,,
to denote
on M, by
Ar = b by Gaussian
is backward
being
algorithm
&,(A)]
of A. The spectral
stable
if
of A [JC~(A) = II AlI 2 II A-‘11 2 = L(
Cholesky
properties
of these
eigenvalue
solving
precise)
is not too large.
product
part
&,,,,(A)
smallest]
(I] . ]IF) are defined
to be
number
[with
The
in [6].
we
[8] showed
decomposition,
outer
x E C”.
2,
discuss
A Q B if B - A is positive
tive definite]
all nonzero
A + A*
=
Hermitian We
[respectively,
Wilkinson condition
) 0) for
of A is
matrices.
A is Hermitian,
If braically the
part
with positive
of positive
r*Ax
of A is
H(A)
Matrices
771
THEORY
was based
(n -
1)
x
elimination
A is positive A)/&,,(
(the Cholesky definite
A) because
and
on the fact that if we partition
(TV -
l)], then
we are left with the (n -
1)
after
the
A is posi-
one step
A as
of the
x (n - 1) matrix
A21 42
A=A,,-
-,
All
which
is also positive
definite
and for which
and Lax(A) which
together
imply
K~( A)
Q K~(
(2)
Q %,m,( A).
A). This fact ahm
the induction
to proceed.
772
FRANK UHLIG, When
A has positive
definite),
the leading
definite
principal
Gaussian elimination
TIN-YAU TAM, AND DAVID
Hermitian
minors
of
and run more
so one would like
efficiently,
elimination
the inequalities generalize
is not necessarily and so one
without pivoting (but there is no guarantee arithmetic).
positive definite.
(but
A are nonzero
be stable in fmite-precision Gaussian
part
without
Algorithms
to determine
will be backward
This is the motivation
for this research.
(1) and (2) to matrices argument
In [4] it was argued Gaussian elimination
with positive
heuristically
precision
definite
Hermitian
solving
Ax = b provided
that the algorithm
conditions
under
stable assuming Our approach
definite
will
structure
that
which H(A)
is
is to generalize
Hermitian
part and then
to these matrices. that
without pivoting,
in which u is machine
positive
can perform
that do not pivot preserve
pivoting
Wilkinson’s
CARLSON
i,
the solution
and c, is a linear function
part. Using this result,
to
Ax = b computed
by
( A + E) 12= b, where
satisfies
of n, when
A has positive
they argued that it is safe not to pivot when
the ratio
1)H + STH- ‘S II2
IIAll, is not large. (It is easy to show that this quantity is at least I.) We make their argument rigorous
and give a sufficient
arithmetic
(without pivoting)
completion
with positive
son’s argument, matrices First
condition
for the
pivots.
Our approach
definite
let us consider
Hermitian
some
part, in particular
the properties
a matrix.
Previous
Hermitian
part [5, 1, 2, 71 has concentrated
interlacing
inequalities
on matrices
for the arguments
this
results
follows
from
the
identity
part to run to
(Theorem
of the Hermitian
with positive
definite
part of the inverse of such
definite
on the properties
of the eigenvalues
of Wilkin2) involving
and submatrices.
of matrices
with positive
ues of AA- l* all have unit modulus.) We start by determining the Hermitian
in finite-precision
Hermitian
of inequalities
part, their inverses
of the properties
Hermitian
research
definite
is based on a generalization
and for this we need a variety
with positive
LU factorization
of a matrix with positive
Hermitian of AA-‘*,
of AA-‘*.
or skewespecially
(The eigenval-
part of the inverse of a matrix. The proof of + Y-l = X-‘(X + Y)Y-’ applied with
X -’
X = A, Y = A*. LEMMA 1. Let A have positive definite Hermitian part, and let H = H(A) and S = S(A). Then A is invertible and A-’ has positive definite Hermitian part given by
A-’
H(A-‘)
=
+
2
A-‘*
= (H + s*H-‘s)-~,
AUBURN
1990 CONFERENCE
ON MATRIX
THEORY
and we have the inequalities IIA-‘112 Define
IIH-‘II,
G
the functions
llAll,< llff+S*H-‘Sl12.
ad
j and K~ on the cone of positive
definite
matrices
by
( A-ly‘-l*)jl~H+s*H-ls
f(A)=
and K+)
where
and S = S(A).
H = H(A)
Many results matrices
involving
with positive
that K~( A) =
=
K~(
1IH+S*H-‘SI12IIH-‘ll,,
Notice
the condition deftnite just as
A-‘),
is convex with respect
that
Hermitian K~(
A) =
A) =
K~(
numbers
part when K~(
to the partial order
A) if A is positive
K~(
of positive K
definite
is replaced
matrices by
K*.
definite. hold for
Notice ako
A-‘). Th e next result, which states that f ,< , is crucial, and numerous inequalities
follow from it (we only state a few here).
2.
THEOREM
order
Q
Let
f be
defined
by (5). Then f is convex with respect
. That is, for any A,, As EM,
with positive definite Hertnitian
to the partial part and any
t E [O, 11,
f(% + (1 - +z) Furthermore, partitioned
suppose
that A, B EM,,
have positive definite Hermitian
(7) part and A is
as
A=
and bt
Q tf( 4) + (1 - t)f( As).
f( A)
with
be partitioned
A,, E Mn-k,
in the same way. Then
1.
Ijf(A22
2.
f(4,
3.
f(h)
Gf(Ah,
4.
aH(A)
2 II Allzll A-‘112
5.
q,(
6. 7.
K”(A)
- ~zlK&)((,
@
A,, E Mk,
4422) Gf(4,
~ljf(A)22112~ ef(A)zz,
= KZ(A) > 1,
tA + (1 - t) B) ~mmax{~~(A),K~(B)}f~anytE[O,1], 2 KH(&
@ AZ,) 2 KH(&)~
KH(A) 2 KH(AZZ - Az,4?‘4,).
Notice that if A has positive of A then, combining
definite Hermitian
4 and 6, we have
K~(
B) Q
part and B is a principal submatrix K~(
B) Q K”(A).
That is, we have a
774
FRANK UHLIC,
bound
on the e-norm
positive definite
condition
Hermitian
number
TIN-YAU TAM, AND DAVID
of any principal
submatrix
CARLSON
of a matrix with
part.
We also prove the following perturbation
result for the function
f by a straightfor-
ward argument.
Let A = H + S have positive definite Hermitian part, and let E be such
LEMMAS. that
[If(A) -f(A+E)II
~~ll~ll,Il~-‘IlzlI~+~*~-‘~l~~=~ll~ll~~~(~)~
Note that if we restrict stronger than (9):
A and E to be Hermitian,
IIf(A+E) -.@)]I,= regardless
of the value of
purposes,
since
A). However,
K~(
our results
in Theorem
Now we consider algorithm
without
the backward
pivoting
part using finite-precision for a complete
arithmetic.
discussion.
matrices
to Hermitian
LU factorization definite
Hermitian
a rigorous
see [3]
are real in our final result.
part of A, is the same as the Hermitian
error of the LU factorization
solution to Ax = b computed
for our matrices,
are many reasons to avoid pivoting;
already proved above and an induction
This in turn can be used to derive
is
in [S] (up to a constant).
of the outer-product
to a matrix with positive
There
[In this case, (A + AT)/2, the symmetric
R =
restricted
We will assume that all matrices
A.] Using the inequalities bound on the backward
the bound (9) is quite satisfactory
stability
when applied
which
IIEII,,
4, when
reduce to the bounds proved for Hermitian
then we have a result
(9)
argument,
part of
we have a
of A with H(A) positive definite.
bound
by Gaussian elimination
on the backward without pivoting.
error
in the
Given a matrix
[ bjj]. we define I B I = [ I bij I]. THEOREMS.
and S = S(A).
Let A E M,(R) have positive dejkzite Hermitian part, and let H = H( A)
Then L and V, the exact LU factors
IIlL Let u be machine
precision.
of A, satisfy
IUl(I,~nllH+STH-‘SII~.
lf
%n3/2~H(
A)” < 1,
(10)
AUBURN
1990 CONFERENCE
775
ON MATRIX THEORY
then the LU factorization algorithm runs to completion and the computed factors i and fi satisfy IIifi-
7un3”]]H+
All,<
S*H-lS]Js.
(12)
Block LU factorization algorithms (see, e.g., [3, Algorithms 3.2.5, 3.2.61) typically will not produce exactly the same computed LU factorization as scalar algorithms (e.g., [3, Algorithm conclusions, x2(B)
3.2.4]),
but
one
with different
may
constants
< xn( A) for any submatrix
expect
the error
in (11)
analysis to produce
and (12),
since
we have
B of a positive definite matrix
similar
shown
that
A.
REFERENCES K. Fan, On real matrices
with positive definite symmetric
M&linear
(1973).
Algebra 1: l-4
K. Fan, On strictly dissipative matrices, G. Golub and C. Van Loan, Baltimore,
component.
Linear Algebra AppZ. 9:223-241
Linear and (1974).
2nd ed., Johns Hopkins U.P.,
Matrix Computations,
1989.
G. H. Golub and C. Van Loan, Unsymmetric
positive definite linear systems, Linear
(1979).
Algebra AppZ. 28:85-97
C. R. Johnson, An inequality for matrices whose symmetric Linear Algebra AppZ. 6:13-18 R. Math&,
Matrices
with positive definite Hermitian
systems, SIAM J. Matrix Anal.
Appl., to appear.
R. C. Thompson,
matrices
11:255-269
Dissipative
part is positive definite,
(1973).
and related
part: Inequalities results,
and linear
Linear Algebra
AppZ.
(1975).
J. H. Wilkinson, International
A priori error
analysis of algebraic
Congress of Mathematicians,
WARING’S
PROBLEM
FOR
by BORIS
REICHSTEIN4’
processes,
in Proceedings
of
1968, pp. 629-640.
SMOOTH
CUBIC
Let (p be a cubic form, i.e., a homogeneous
CURVES
polynomial of degree 3. We would like
to determine the smallest integer k such that (p can be expressed as a sum of cubes of k linear forms and find all corresponding
representations
tions. This question is known as Waring’s proposed belonging
an algorithm
that allows one, for some values of k and for cubic
to a wide class of forms
42Deparhnent 20064.
that we call Waring presenta-
problem for cubic forms. In [l, 21 we have
in n variables,
to find almost
of Mathematics, The Catholic University of America,
forms
all the Waring
Washington,
D.C.
776
FRANK UHLIG,
presentations algorithm, almost
or to prove however,
all Waring
presentations
form
form
written
in canonical
linear
forms
form
appear
by cube
root
two-dimensional. investigate
In this case
coordinates in Waring
is unique
that
the cubic
presentations.
formulas
It turns
vanishes;
the variety
X,
obtained
by methods
of algebraic
that, opposite to the case of quadratic forms, X,
and
X,
work
of the we find
corresponding
(p smooth.
For the
multiplication
of Waring
inspired
geometry. is irreducible
of
k = 3 if and only if
k = 4. In the first case
of terms
in this
work
for the coefficients
out that
otherwise
(up to reordering
formulas
curve
we also call the form
of 1). In the second case the variety The
Some of the final steps
n > 3. In the present
we find explicit
[3, p. 3021 of the curve
presentation
exists.
only to the case for n = 3 provided
(p is smooth.
that
the j-invariant Waring
that no such presentation
are applicable
to the given
TIN-YAU TAM, AND DAVID CARLSON
Zinovy
the
of each
presentations
is
Reichstein
to
It has been established and irrational [4].
In order to derive the desired formulas we first exploit those steps of the algorithm in [1, 21 that are applicable to the case n = 3. The algorithm requires form (p = x: + 3x3(x:
Any smooth cubic fom
LEMMA 1. appropriate
linear
transfotmation
cp in three variables
of variables,
Let cp(zl, z2, z3) be an arbitrary
Proof. linear
cp to be in the
+ ~22) + (pp(x1, x2) where p2 is a cubic form in xl and x2.
can be written,
after an
as
cubic
form.
It is known
that there
exists a
T, : ( zlr z2, z3) + { y,, yz, y.J such that
transformation
$0( y1, yz> Y3) = Y? + Y$ + Y33+ 3OYl YzY31
(2)
0 # 0 [3, p. 2931. Let
xl
Tz: { ~1, YZ~ ~3) -+
+
ix,
J;;’
x1
J;;’
-
ix, ~3 I
Then the form Q becomes T,T,
maps
the arbitrary
as in (1) with p = 2/ 0.
form
Since the inverse transformation find all Waring presentations In order prescribes
T; ‘T;’
n
maps (1) into (p(zl, z2, .z3), it suffices to
for the form (1).
to find a Waring
introducing
Thus, the linear transformation
cp into the form (1).
presentation
the following matrices:
of the form (1) the algorithm
in [1, 21
AUBURN These
1990
matrices
CONFERENCE commute
if and
expressible
as a sum of cubes
~1 = z/2,
then
gl=
ON MATRIX only
if $
c3 = 8. Thus,
to [2], these
rp(%~,A,)
vectors
the Waring
5 j=l
: g3jkTl
gli,
g,i,.
. . , g3i are coordinates
(
-fix,+
Therefore,
for the form
form
g,=(-$-
$,lJ1.
presentation
(gljrl
+
g2jx2
+
g3jx3)3.
(4)
dj
of the vector
vGx,+2x,
3
gi, i = 1,2,3.
- fix,
-
&x2
Thus,
+ 2x3
3
+ 2% (2) with
(2) is
of Ds are
define
=
the
forms if and only if o = 0 or o3 = 8.43 If
{l,o,2E]t, g,=(-$.$Ji’.
According
where
= $. Then
of three linear
the eigenvectors
777
THEORY
I u = 2 we obtain
i the following
2%
(5) I
representation
as a sum
of three cubes:
(
(l-iv5)yl+(1+iv%)y2+2y3 2fi
i (l+ifi)yr+(l-iv5)y2+2y3 2%
13 +
1.3
430f course, this result is consistent with the fact that the j-invariant of the form (2) is a3(8 - a3)/(03 + 1)3 [3, p. 3021 and therefore the form (2) with (r = 0 is equivalent to the forms (2) with o3 = 8.
778
FRANK UHLIG,
TIN-YAU TAM, AND DAVID
CARLSON
From now on we assume
u # 0,
Under these conditions cannot be expressed
a3 - 8 # 0,
the matrices
2pa -
(3) do not commute
1 # 0.
(6)
and the form (2) as well as (1)
as a sum of cubes of three forms. In order to express
cubes of four linear forms we introduce,
in accordance
with the algorithm
c as a sum of in [l, 21, the
following matrices:
fii=
; 1P
-p
0
r
0
0IP
0
r
OS
p
Here
p, 9, r, s, and t are arbitrary
show that the commutator
0
I
El,=
)
i
complex
[Dir Da] vanishes
2~’ + r2 9=
1
P
s=
parameters.
-
p(2$
-
2,s
P(2P2
For the further
calculations
this expression the characteristic
polynomial
4(A)
=x4+
-
ci = -r{2P4
c3 = P{2Pz
-
-
1)
- 1
I)
(8)
.
for s only. After substituting
c,x3 + c2x2 + c,x + Co),
1
+ ( p2 + r2 + 3)9
-
+ 2r2
( p2 - 3r2
calculations
for
- N) of 6,:
( p” - r2 + l)$
cs = - {2P4 + (2p2
I
Straightforward
where co = - (2~” -
(7)
in (7) we obtain the following expression
4(X) = det(fir
‘i2$
t
- 3)
we shall need the expression
from (8) into the matrix 6,
r 9
- p2 + 3r2
- p2 + 3rs
rr(@
0
0 I 100’ 0 9
if and only if
’
r=
-P
-n 0 r
+ r”),
- 2( p2 - 2r2
+ 1)~~ -
+ 1)).
+ l)},
( p2 + r2 + l)},
(9)
AUBURN
1990 CONFERENCE
If h,, &, Xs, X, are eigenvalues of d,, corresponding
gi=
eigenvectors
779
ON MATRIX THEORY
they are distinct for almost all p and r. The
are
{-p&(P+X,),r(p&-
+CLjtT
l),-P(~+Ai),-A~+~(S+l)
g+
i = 1,2,3,4.
Each of the eigenvalues
Xi is a function of the parameters
(o(% X2-Tx3) =
5
:
( gljxl
+
(II)
p and r. The formula
g2jx2 +
(12)
g3jx3)3p
j= ’ g3jkGldj where
{ gii, g2i, g3i, gdi} are coordinates
find Waring presentations the coordinates
of the vector
that we have found so far, none of the coefficients
vanishes. Now we will find Waring presentations linear forms the variable the scalar product
(
x(
x3 does not appear explicitly.
of x3
that there
exists no
of x3; otherwise,
Let again p( xi, x2, x3)
a new cubic form
x2>
x3)
=
v(
x1,
x2,
x3)
-
p and r are arbitrary complex parameters,
defined later. We compute The commutator
Notice
, ) defmed in [l, 21 would be degenerate.
Xl,
of
of the form (1) such that in one of the
of (1) where more than one linear form is independent
be the form (1). Introduce
where
allows us to
of gi from (11) into (12), we obtain almost all Waring presentations
the form (1). In all Waring presentations
representation
gi (i = 1,2,3,4),
for almost all values of p and r (see [I, 21). Thus, substituting
the matrices
4(
PXl + fl2),,
(13)
and q is a function of p and r to be
(3) for the form (13) and their commutator.
vanishes if and only if
2/P4=
1
PP( P’ - 3r2)
If p and r are arbitrary parameters
’
(14
and q is as in (14), the form (13) is expressible
as a sum of cubes of three linear forms. Let s = p/r
be a nonhomogeneous
parameter.
780
FRANK UHLIG,
TIN-YAU TAM, AND DAVID CARLSON
The matrix D, associated with the form (13)-(14)
[see (3)] now becomes
_ /q ss + 3) - ss P(s2 D, =
- 3) _ p2(s2 -1)-l
- +;:31/
0
P( s2 - 3) 1
I Its characteristic
0
polynomial 4(X) = det(N
01
- Dr) is
(s2 + 1) 2p
f&(X) =x3+
-
1
x2 +
P(S2 - 3)
cl”( ss + 1) - 2( s2 - 2) x _
cl”( s2 -
ss - 3
1) -
1
(16)
q-3)
If h,, &,, A3 are the roots of the last polynomial, the linearly independent s) eigenvectors
(for almost all
of the matrix (15) are
g, = { Xi[ /.&(3 - S2)Xi + $(l
- s”) + 11,
XiS(2$
(i = 1,2,3).
(15)
.
-
l),p(3
- S2)hi + p2(1 - s”) + l}t
As above, we obtain the desired set of representations
of the form (1):
+&-3/4x,x,2=
SXl + x2) 3+
c
j=l
1
g3j
(
g12j +
) gij
+
( gljrl
+
g2jx2
+
g3jx3)3.
dj
From the results of [4] it follows that the roots of the polynomial (9)-(10) made rational by a substitution of variables
can be
p and r if and only if /.I~+ 4 = 0 and hence
u3 + 1 = 0. In this case the cubic form (8) [and therefore
(9)] is a product
of three
AUBURN
1990 CONFERENCE
ON MATRIX
THEORY
781
forms. If (r = - 1, we have
Y?+Y23+Y33-3Y1Y2Y3=
(Yl + Yz + YB)(%Yl
+ EZYZ + YB)(%Yl
+ ElYZ + Y3)p (17)
where cl and Ed are cube roots of 1:
The form (17) can be expressed
as a sum of cubes of four linear forms as follows:
It is trivial to obtain the similar formulas for the form (8) with (I = E~ and u = Ed.
REFERENCES B. Reichstein,
An algorithm
to express
a cubic form as a sum of cubes of linear
forms, in Current Trends in Matrix Theory, Proceedings of the Third Auburn Matrix
Theory Confwence, North Holland, 1987, pp. 273-283. B. Reichstein, appear.
On Waring’s
problem
for cubic
forms,
Linear
Algebra A&.,
to
782
FRANK and H. Knorrer,
UHLIG,
3
E. Brieskom
4
B. Reichstein, Z. Reichstein, On Math. -I., submitted for publication.
TIN-YAU
Plane Algebraic Waring
TAM, AND DAVID
Curoes,
Birkhsuser,
presentations
CARLSON
1986,
of plane
pp.
cubits,
1-721. Michigan
OUTPUT FEEDBACK CONTROL OF LINEAR REPETITIVE PROCESSES-A 2D POLYNOMIAL MATRIX APPROACH by
E. ROGERS44 and
Repetitive,
or multipass,
can be illustrated piece,
H. OWENS45
D.
involved
processes
by considering is processed the output,
forcing function
on, and hence
length a is constant, hence
operations
of sweeps,
or passes,
contributes
to, Yk+l(t),
processing
structural problems
and Owens
of the
over a by Y,(t),
processes
[Rogers and Smyth length
also
(1989)],
to the current M, or simply
The
termed unit-memory.
essential
studies
system. obviously appropriate
Roesser
example,
image-
model
[Rogers
Such processes and
process
are
unit-memory
can be regarded
bench-mining
systems
M > 1 passes which contribute termed
nonunit-memory
in the special
case
of
of M = 1.
as the natural generalization
results Further, totally
control
problem
Such behavior
on actual
is easily
processes.
from simulation it is clear
stability
and
Rogers
in the special from
control
[see,
for
example,
the
of its
control
44Department of Aeronautics 45SchooI of Engineering,
action in
methodology
and Astronautics,
University
studies Smyth
is required {Yk}kal.
is required, linear-dynamics
in Rogers
University
of Exeter, U.K.
and
is the
increase
possible
in amplitude and observed (1989)
contains
case of one type of bench-mining
appearing
treatment
process
which
in simulation
example,
appropriate
feature
analysis
a repetitive
generated
For
studies
that
undesirable
for
{ Yk}k 5 1 of oscillations
sequence
aspect is not necessarily feedback-based. A rigorous stability theory for the constant-pass developed
operations,
and 2D
state-space
so-called
it is the previous
pass profile.
unique
in the output
extensive
subclasses
counterpart.
from pass to pass. in field
certain
by the
exist-for
where
nonunit-memory
Hence a nonunit-memory unit-memory presence
described
0 Q t < a,
(1990)].
Repetitive directly
type
exist between
a
finite pass
acts as a forcing function on, and
0 6 t < a, k > 0, and is therefore
similarities
In
To introduce
Industrial examples include long-wall coal cutting and certain metal-rolling and strong
tool.
pass acts as a
suppose that the necessarily Y,(t)
which
or work-
of the processing
to, the next pass profile.
is one where
process
action
the material,
on the current
and denote the pass profile generated
a repetitive
contributes
by a recursive where
or pass profile, produced
[Rogers and Owens (1990)],
definition
k > 0. Then
characterized
by a series
such operations, formal
are
machining
and
Owens
to prevent
this
In particular, where
the
an latter
case has been (1990)]
using
of Southampton, U.K.
an
AUBURN 1990 CONFERENCE
ON MATRIX THEORY
783
abstract model formulated in functional-analysis terms which includes almost all known examples, or subclasses, as special cases. This has shown that two distinct concepts are required. These are termed asymptotic stability and stability along the pass; the former is a necessary condition for the latter, which is clearly required for all practical purposes. In effect, stability along the pass demands that, given well-defined inputs or driving terms, { Yk)k a r converges strongly to a steady, or limit, profile irrespective of the pass length a. The results of applying this abstract theory to a wide range of special cases have been reported [Rogers and Owens (1989a, b)], and it is known that the resulting conditions have well-defined physical interpretations. In particular, the results for the subclass of so-called differential nonunit-memory linear repetitive processes are well known and understood. This subclass includes the bench-mining systems as special cases and has the following state-space model:
y/r+l(t)= ‘X,+1(t) +
5
Djyk+l-j(t)-
j=l
Alternatively, suppose, for simplicity, that X k+r(O) = 0, k 2 0, and Y,_j(t) = 0, 0 Q t < a, 1 < j < M. Then it can be shown [Rogers and Owens (1990)] that (1) has the 2D transfer-function matrix description
Y(s, 2) = G(s, z)U(s, z), where the m
x 1 2D
transfer-function
Go(s)
(2)
matrix G(s, z) is defined by
= C(s& - A)-‘B
(4)
and Gj(s)
= C(s& - A)-‘Bj_,
+ I+,
l
(5)
FRANK UHLIG,
784 This subclass
has clear
structural
TIN-YAU TAM, AND DAVID CARLSON
similarities
to standard
or, in repetitive-systems
language, conoentional linear systems. Suppose that the previous pass terms are deleted from (l), the subscript Then
k + 1 is dropped, and the concept
the result is just the standard
(A, B, C), which is termed has transfer-function B = 0, B,_r
state-space
model
of a pass length is irrelevant. characterized
by the triple
the derioed conoentionaZ linear system in this context,
matrix
G,,(s), i.e. a constituent
= 0, Di = 0, 1 < i # j Q
and
element of G(s, z). Similarly, set
M, drop the subscripts, and ignore the concept of
a pass length. In this case the result is just the standard state-space model characterized by the quadruple
(A,
Bj_l, C,Dj),1 Q j Q M, which is termed the jth associated
conoentional linear system, and has transfer-function constituent The
structural
developing
matrix
Gj(s) of (5), i.e.,
similarities
a comprehensive
summarized control
above
theory
have
motivated
an approach
for (1) based on using directly
extending (where possible) the well-established
conventional
feasible stability tests (asymptotic
pass) based on G,(s)
Q
however,
and Gj(s), 1
to
and/or
linear systems theory.
date, this has yielded computationally remains,
another
element of G(s, z).
To
and along the
M, or their state-space realizations. There still
work to be done,
aspects of the role of G( s, z). In particular,
and this exposition
will focus on other
the following results, conjectures,
and open
questions will be addressed.
THEOREM 1.
Theprocess (1) is stable along the pass if, and only if, the characteris-
tic polynomial p(s, 2) satisfws Res>O,
P(S,2) f 0,
(.zI >l,
(6)
where
p(s,z):=
sl, - A c
-W
Q(z)
and
B(Z)
Proof.
=
5 Bj_lz-j,
Q(z)
j=l
= ,$Djz-j. 1, -
See Rogers and Owens (1991).
Suppose that (1) is embedded in an output-feedback-based unity-negaTHEOREM 2. tive-feedback control scheme dej%ed by
U(s, 2) = K(s, z)e(s,
2) = K(s,
z)[R(s, 2) - Y(s, z)],
(9)
AUBURN
1990 CONFERENCE
where R( s, z) is the reference
signal and K(s, z) has the structure
the open-loop forward-path p,( s, z) respectively.
785
ON MATRIX THEORY
and closed-loop
characteristic
of (3). Further,
polynomials
denote
by p,,(s, z) and
Then
Pc(s, z) =p(s, Z)I> Po(STz)
(10)
-
where the return difffence
matrix T(s, z) is given by
T(s,
z) = Z, + G(s,
z)K(s,
2).
(‘1) n
See Rogers and Owens (1991).
Proof.
DEFINITION. The natural definition of a pole for (1) is a pair of complex numbers (& z^)which satisfy p(s, z) = 0.
CONJECTURE. A pole of (1) has a well-defined physical interpretation used to characterize
stability in a similar manner
which can be
to its conventional
linear-system
counterpart.
OPEN QUESTIONS. 1.
What is the equivalent of the Rosenbrock
system matrix for (l), and how (if at
all) can it be used to define and answer fundamental 2.
systems-theoretic
questions?
How (if at all) can p(s, z) and T(s, z) be used in the development
controller
of efficient
design algorithms?
REFERENCES Rogers, E. and Owens, D. H. 1989a. processes, Rogers,
E. and Owens,
processes
D. H. 1989b.
with non-unit memory,
Rogers, E. and Owens, D. H. 1990. Lecture
Systems, Research E.
Stability analysis for discrete
Stability Analysis for Linear Repetitive Processes, Sci., Springer-Verlag,
D. H.
differential non-unit memory submitted for publication.
to appear.
Modeling and Simulation Studies on Bench Mining
Report, Div. of Dynamics
and Owens,
linear multipass
ZMA J. Math. Control Znfonn. 6(4):399-409.
Notes in Control and Inform.
Rogers, E. and Smyth, K. J. 1989. Rogers,
Axis positivity and the stability of linear multipass
Linear Algebra Appl. 122/123/124:779-796.
1991.
and Control,
An output feedback
linear
repetitive
processes,
Univ. of Strathclyde. based control Linear
theory
Algebra
for
Appl.,
786
FRANK UHLIG, TIN-YAU TAM, AND DAVID CARLSON
MINIMUM
PERMANENTS
ON CERTAIN
DOUBLY
by SEOK-ZUN
SONG46
I.
AND
MINIMIZING
STOCHASTIC
MATRICES
MATRICES
Introduction For a pair ( p,
q) of positive integers, J,, 9 will denote the p x 9 matrix all of whose
entries are 1. Let the set of all n-square doubly stochastic This
is known
to be a polytope
n2-dimensional
Euclidean
Let D = [d,,J
= {X=
with
n! vertices
in the
[x~,~] ~Q,(x~,~=Owheneverd,,~=O).
Euclidean
X E Q(D). Such a matrix
matrices be denoted by Q,. 1)’
matrix, and let
Q(D) is a face of the polytope
finite-dimensional
(n -
space.
be an n x n (0,l)
Q(D)
Then
of dimension
and hence,
Q,,
space, contains a matrix
being a compact
subset of a
A such that per A Q per X for all
A will be called a minimizing matrix on n(D).
Without any doubt, one of the most interesting
and important problems concerning
the face Q(D) is that of determining
the minimum value of the permanent
the set of all minimizing
on it, of which many studies have been done by
several
authors.
determined
For
matrices
example,
the minimum
Knopp
and Sinkhom
permanents
on Q( Dl),
[6], Mint
function and
[7], and Brualdi
[l]
Q( Dz), and Q( D3), respectively,
where
D,
=
_“_-~~_::-_-l-, ”
*..
D,=
1,n
1
0
0
. . .
0
1
. . .
0
1
10
. . .
1
1
. . .
1
. . .
0
1
..------
1
1’
Jn-z,n
0
1
D,=
i 1
.
1
.
1
. .:’
1
I.
Friedland [4], Hwang [5], Foregger [3], and Chang [2] also determined permanents on certain faces (see [9]).
the minimum
4sDepartment of Mathematics, Cheju National University, Cheju 690-756, Republic of Korea, and Department of Mathematics and Statistics, Utah State University, Logan, Utah 84322-3900.
AUBURN 1990 CONFERENCE
787
ON MATRIX THEORY
In [I], Brualdi determined n( W,), where
the minimum
permanent
and minimizing
1
0
0
..:
1
0
1
0
0
a**
0
1
matrix on
We wanted to extend this result from W,, to V,,,,nr defined as
In [ll], we determined the’ minimum permanent and minimizing matrices on Q(V,,J for arbitrary n. In [lo], we determined them on Q(V,,s) for m = 2 and m 3 5, but not for m = 3,4. In [12], we determined them on Q(V,,,) for all m 2 3 by a method somewhat different from that in [lo]. In [12], we also determined them for W,,” and
W,, .(O), where I
02,”
I m,m 1 I 12,”
WIn.” =
----L_-__
i
1 7l.m ,’
I
1 ’
An-2.m -----
%“(q
I Om-2,n 1------
02 _--F_~___~“_
=
[
I n, m
I
12 I
1
for n ) 2, m ) 3. LEMMA 1 (Foregger
[3]).
Let D = [d, j] be an n x n fdy
indecomposable
matrix, and A = [ ai, j] be a minimizing m&-ix on Il( D). Then A is fdly
(0,l)
indecomposable,
and for (i, j) such that di, j = 1,
per A( i 1j) = per A
if
ai, j > 0,
per A( i 1j) > per A
if
ai,j = 0.
LEMMA 2 (Mint [7J). [d,, . . . >d,],
and if&
If A = [al,. . . , a,] is a minimizing matrix on Q(D), D =
= d2, then
per[cual+/3a2,pa,+ora2,a3,...,a,] fOranyol,@>Owithcu+j3=1.
=perA
788
FRANK UHLIG, TIN-YAU TAM, AND DAVID CARLSON We will call this the aweraging method when c~ = /_?= i.
THEOREM 1. For m > 3, let
Then a minimizing matrix form A on Q( V,,,,3) is
(I.‘) and the minimum permanent is mlam-‘[(m
- l)mb4 + 2maxb2+
P-2)
?a”],
where mu = 1 - 3 b, x = 1 - mb, and b is a real root of 27 + m
m2+ 6m+20
21 - ?. m
9 3 +m” m3
b4+
i
b-$=0.
(1.3)
THEOREM 2. For m p 2, n 2 3, let
I Om-2,n
Wm,Il=
/ m,m 1
I --- L__*Ln__ 2, i J“,rn ,’
Then the minimum permanent un O(W,,,,) m! &c-
1 .
is
4(n - l)(n g+l
- Z),-’
(2.1)
AUBURN 1990 CONFERENCE
ON MATRIX THEORY
789
fwna4,and
(m - l)!
5b + 6mb2
2mmm2
fm n
(2.2)
= 3, where b is the unique real root of the equutiun
llm2b3
- 16mb’ + 9b - 2 = 0. m
(2.3)
We remark that the matrix W,,” is cohesive and not barycentric for m 2 2 and n ) 4. (For definitions, see [l].) By the averaging method of Lemma 2 and the proof of Theorem 2, we obtain the following result about one of the faces of n( W,. J: COROLLARY3.
For m 2 2, n > 3, let
Then the minimum permunent on Q( W,,,, *(O)) is the same as (2.1) in Theorem 2, which occurs at the barycenter b(W,,,, JO)), where the batycenter of Q(D) is giuen by b(D) = P, and the summation extends over the set of all permutation matrices P (llP~D)&<, with P < D, and per D is their number. REFERENCES R. A. Brualdi, An interesting face of the polytope of doubly stochastic matrices, Linear and Multilinear Algebra 17:5-18 (1985). D. K. Chang, Minimum permanents of doubly stochastic matrices with one fixed entry, Linear and M&linear Algebra 15:313-317 (1984). T. H. Foregger, On the minimum value of the permanent of a nearly decomposable doubly stochastic matrix, Linear Algebra Appl. 32:75-85 (1980). S. Friedland, A proof of a generalized van der Waerden conjecture on permanents, Linear and Multilinear Algebra 11:107-120 (1982). S. G. Hwang, Minimum permanent on faces of staircase type of the polytope of doubly stochastic matrices, Linear and M&linear Algebra 18:271-306 (1985). P. Knopp and R. Sinkhorn, Minimum permanents of doubly stochastic matrices with at least one zero entry, Linear and Multilinear Algebra 11:351-355 (1982). H. Mint, Minimum permanents of doubly stochastic matrices with prescribed zero entries, Linear and Multilinear Algebra 15:225-243 (1984). H. Mint, Permunents, Encyclopedia Appl. 6, Addison-Wesley, 1978.
FRANK UHLIG,
790
TIN-YAU TAM, AND DAVID CARLSON
9
H. Mint, Theory of permanents
1982-1985,
10
(1987). S. Song, Minimum permanents
on certain faces of matrices containing an identity
submatrix, 11
Linear Algebra Appl. 108:263-280
S. Song,
Minimum
permanents
Algebra AppZ. 143:49-56 12
Linear Multilinear Algebra 21:109-148
(1988).
on certain
doubly
stochastic
matrices,
Linear
(1991).
S. Song, Minimum permanents
on certain
doubly stochastic
matrices
II, Linear
Algebra Appl., to appear.
ON RATIONAL by EIVIND
1.
n
MATRICES
and BOONCHAI
K. STENSHOLT4’
and Observations
Let n, t, x be integers,
1.
DEFINITION x
STOCHASTIC
STENSHOLT4’
De$nitions
set of n
DOUBLY
matrices
with integer
entries
n > 0, t > 0, r ) 0, and N,(t, x) be the from
(0, 1, . . . , t} such that all row and
column sums equal r. Let M,,(t) be given by
iv,(t)
The cardinahty
A
of a set S is denoted
COMBINATOBIAL
owners
hold
u
= N”&O)
Aqt,1)
each,
none
0..
u N&q.
) S (.
In each
INTERPRETATION.
r shares
u
more
than
of n companies
t shares
are
x shares;
in any company.
n
A matrix
(aij) E N,(t, x) specifies a possible distribution of shares, owner i holding aij shares in company j. The special problem Anand, matrices
Dumir,
of determining
and Gupta
[l];
in NJ r, r) are sometimes
indicates additional requirements the entry set is {1,2,.
47Norwegian School
( NJ r, r) ] was studied by Mano [14] and
they denote
] NJ X, r) ( as H(n, r)
called magic squares,
for r = x. The
although usually this term
(such as that the two diagonal sums also equal r and
. , n2}; see [2]).
of Economics
481nstitute of Marine Research,
P.O.
and Business Administration, 5035 Bergen, Box 1870,
5024 Bergen,
Norway.
Norway.
AUBURN 1990 CONFERENCE Several properties
791
ON MATRIX THEORY
of these matrix sets follow from the definition:
N,( t, x) = 0
if
t
(1.1)
N”( t - I, x) c N”( t, x)
if
m -‘
(1.2)
N,( t - 1, x) = N,( t, x)
if
x+l
(1.3)
IN,(t,x)l=IN,(t,nt-x)1. [For (1.4) consider
(1.4)
the l-l map (ajj) * (bij) where aij + bij = t for all i,j.]
NOTATION. Let Q, be the set of doubly stochastic n x n matrices 31, and aQ, its interior and boundary in the relative topology of its &ne span W of dimension m = (n - 1)‘. Let r,(x) be the lattice of n x n matrices with entries zx-I, z E Z, and C,,(e) the n2-dimensional cube of n x n matrices with entries in [O, e]. J, denotes the n x n matrix where all entries equal 1. The set 0, is the intersection of W and the nonnegative orthant of @“‘. Its geometry was studied in [4]. From Definition 1 it follows that for x 2 1,
AEN,,(t,
x)
ifandonlyif
n n,.
n C”(K’)
x-~AEI’,(x)
(1.5)
By (1.5) NJ t, r) splits up as follows if x >, 1: N,(t,
TX)= I$( t, x) U aN,( t, x) (disjoint union),
(1.6)
where &(t,
x) = {AEN,@,
and
x)(x-‘Ad,}
aN,(t,
x) = {AEN+,
x)/x-‘A&Q,).
Let (aij) E NJ t, r), x > 1. a62, consists of those matrices in Q, which are limits for convergent sequences in W \ Q,; hence (aij)
Efin(t,X)
(aij)EaN”(t,
X)
ifand only if
aij > 0
for all i, j,
(1.7)
ifandonlyif
aij=
forsome
(I .8)
0
i, j.
From this it follows that
qt,
x) = 0
&(t* n) = {k) AsN,(t
-
1, x - n) ifandonlyif
if
l
when A +],,Efi,,(t,
(1.9) (1.10)
t>l,
x)
when
x > n.
(1.11)
FRANK
792 We notice the following
UHLIG,
consequence
TIN-YAU TAM, AND DAVID
of these observations:
IN,(x-n,x-n)I=IN,(x-1,x-n)I
and
CARLSON
By (1.3) and (1.11)
IN,(x-1,x-n)(=l~“(x,x)(; (1.12a)
hence
p,(x-n,r-n)l=p”(x,x)I, 2.
Triangulations
Proper
A triangulation of any polytope 62 is a finite set y= {T,, Ts, . . . , T,} all faces of simplices in ? are in 9-, n = T, U T, U *. . U T,, and
DEFINITION2. of simplices
where
if Tj tl Tk # 0,
moreover,
(1.12b)
x>n.
said to be proper
then Tj 17 Tk is a face of both Tj and Tk. The triangulation
if all corners
of each Tj belong to the corner
is
set of fl.
THEOREM(Fuglede [8], Brbndsted [S]). Let Q be a d-dimensional conuex polytope with corners p,, p,, . . . , p,. Then there exists a proper triangulation of Cl. Birkhoffs matrices
permutation taining r,(x)
theorem
for corners. matrices
x- ’ * ( aij).
[3] states
that
For a proper
51, is a convex
triangulation,
that are comers
in the unique
( N,,(cc,x) 1 is determined
polytope
(aid) E
with the permutation
N,(t, x) is a unique sum of the
lowest-dimensional
by counting
the number
simplex
con-
of points from
in each simplex. So, let A be a simplex
with the permutation
matrices
PO,P,, . . . , Pd for comers.
The points
PO+ CkiX-‘(Pi form a sublattice A of r,(x).
- PO),
kiEZ7
l
If d Q 3, it is easy to see that A = r,,(x) whether
A actually
(2.1) fl D, D being the
alline
span of A. It is not known
can be a proper
r,(x)
tl D for (high) values of d, but the possibility must be considered.
sublattice
of
Since the two
lattices have the same affine span, A is of finite index, say s, in I’,,(x) il D. From the kth coset we pick the unique representative
Qke
{p.
+
=j+x-‘(Pi
Qk such that
- P,,)jO < ri G 1, 1 G i 6 +
thus r,(x)nD=(~,+n)~(~,+a)u~~~u(~,+A).
(2.2)
AUBURN Let
1990 CONFERENCE
Qi + A = A, i.e.
793
ON MATRIX THEORY
Qi = x-‘[(x
- d)Pa + P, + .*.
+Z’d]. Each
x-l*
(aij)~r,(x)
fl n, is counted in the simplex of smallest dimension which contains the point, i.e., we count the inner lattice points in each simplex. Using
we count layer by layer. Here
y = z ‘when we count the points of (Qr + A) tl A; the
points is in a face of A and is not counted.
Qk = Pa + -&ix-1(P,
- P,,),
Now, write
0 < q.‘kiQ 1,
(2.3)
and let bi E Z be such that b, Q Crki < b, + 1. Then, for 1 < k < s, we have 0 < b, < b, = d, and the number of points from Qk + A in the interior H of A is
(2.4) thus (r,(x)nAI=
2
‘+d,‘-,,i,
for k>l.
d=b,>b,
(2.5)
k=l
Stanley [17] has proved the following result, originally conjectured
in [l]:
THEOREM (i) ] N,( x, r) ] is a poZytaomiaZin r of degree (n -
l)‘,
x )
1.
When x is allowed to assume also nonpositive values, (ii) the polynomial has n - 1 zeros: IN”(_l,
-
l)/
=
**.
=IN,(l
- n,l
- n)I = 0,
and (iii) it has the following symmetry
1%(-n-x9
-
A simple proof of (i) and (ii).
property:
n-x)/ = (-l)“-‘(~,(~, The exi$ence
of proper
x)I, triangulations
and (2.5)
show that ]N,(r,x)l, la&(x,x)], and I&,(x,x)] = INn(x,x)) - l?JN,,(x,~)l all are polynomials for x > 0. So (i) holds, and (ii) follows from (1.9) and (1.12) with x=1,2 ,..., n-l. n
FRANK UHLIG,
794 (iii) is a special
case of Ehrhart’s
TIN-YAU TAM, AND DAVID
reciprocity
deeper nature than (i) and (ii); see Remark
theorem
CARLSON
[7]. It still seems to be of a
2. A proof in line with this account
is in [12];
see also [6, 7, 15, 171.
REMARK 1. If s = 1 for a simplex of maximal dimension s = 1 for each
of its face simplices
too.
Since
tributed,
the volume of a maximal-dimensional
construct
maximal-dimensional
such that the number increasing integers;
i. [Then hence
simplices
of zero
(n -
points
1)2, then clearly are uniformly
simplex is proportional
in
P, + P, + * . . + Pi decreases
PO, P,, strictly
fl W and x:zi = 1 imply that the
X.ziPj = (aij) Ed,
dis-
to s. It is easy to
such that s = 1 in (2.5); just choose
entries
. with
xzi are
( uij) E A.]
PROBLEM. Are there maximal-dimensional are there
the lattice
cases with
s > l? If so, are there
simplices proper
with different
triangulations
volumes,
which
i.e.,
avoid these
cases? Let the polynomial
1N,,(x, x) ) be expressed as follows:
(2.6) A proper
triangulation
simplices.
The symmetry
a small number computer. With
The
more
functions
s = 1 for all simplices
(iii) allows the polynomials
of values; sequences
refined
completed
with
for
n < 5 sufficiently
{ ( N,( x, x) ( },
techniques
must
many values
r = 0, 1,2,
and a computer, a,
a,_i
ai i-dimensional
have been
from
found
by
. . appear in [16] for n = 3,4.
Jackson
the case n = 6 too. They also gave equivalent [17]. Thus one obtains
contain
) NJ x, x) 1 to be determined
and van Rees
[lo]
have
results in terms of generating
a, for n = 3;4;5:
.
3 12 19 15 6; 352 2464 7544 4.718075
2,905,658,575 14,062,951 These
13,232
51.898825
14,620
2,463,775,850 1,784,345
coefficients
indicate
complex.
In a proper
with
common);
9 vertices
1468 258 24; 1706.729525
2584.561500
770,476,155
280,134,105
74,580,465
120. triangulations,
ai > 0 and u,_r
with ai i-dimensional
= [n + $(n” - 3n + 2)]a,.
simplices, So the sim-
seem, on average, to have n facets (faces of dimension remaining n2 - 3n + 2 are walls in the triangulation
triangulation of valency
of Qs, the 3-dimensional
simplices
4 (adjacency
a 2-dimensional
is to have
similarly Iah+
indicates
6090
that proper
plices of maximal dimension m - 1) on aa,,while the
4945
811.572625 1,584,408,615
140,740
exist and are very regular:
graph
10,532
262.803150
x)1 =IN&,
a graph with 1408 vertices
x)] -IN+
- 4, x - 4))
of valency 9 on aQ,.
on an,
form a face
in
AUBURN 1990 CONFERENCE
795
ON MATRIX THEORY
PROBLEM. What is true for general
n?
Factorized forms of (2.6) for n = 3,4 illustrate (ii): 8- ‘( x + I)( x + 2)[( x + l)( x + 2) + 21 and 11,340-l( x + l)( x + 2)(x + 3)[11( x + 2)6 + 23(x + 2)4 + 128(x + 2)’ + 3061. The first is (33) in [l], and MacMahon [13, Section 4071 gave the form
3(x:3)
REMARK2.
+ (x:2).
By (1.6), (1.12), and part (iii) of Stanley’s theorem
P”(X, x)1=lN,(--)I -I NJ r -
we have
.,x-.)I=IN,(x,x)l+(-l)“lN,(-x,-x)1;
hence (aN,(O,O)j
= [1 + (-1)“]
*IN,,(O,O)l.
(2.7)
For x = 0, (2.5) becomes
i
-;I=(_l)d
Summing for the simplices in 62, and an,, we see that ] N,,(O,0) ] and ] aNJO, 0) ] are the Euler characteristics of 0, and do,, i.e. of the ball and sphere of dimensions m and m - 1. Hence (2.7) is a topological relation which follows from (iii); for this reason it would be interesting to prove (iii) as simply as (i) and (ii). [From topology ] N,(O, 0) ] = 1, i.e., the polynomial gives the correct value also for x = 0.1
3.
Error-Correcting
Codes Related to fl,
The set M,,(t) from Definition 1 is a code with alphabet {0, 1,2, . . . , t}. If (aij) E M,(t), n >, 3, is received with an error in a single entry, this is located by means of the deviating row and column sums, and uniquely corrected. The Hamming distance is at least 3 if n = 3, at least 4 if n is at least 4. This code may well be used as an identification code like the ISBN book code [ll], the universal product bar code UPC [18], or the codes for bank account numbers. For general information see [9]. These well-known codes detect errors only, and a code which also corrects errors may be a worthwhile alternative. The t&,-codes proposed here can be organized in different ways, e.g. according to the row and column sum x, and according to the simplex in a given proper triangulation to which a codeword is associated. Thus codewords may be assigned systematically and without repetitions, e.g. by different local authorities, each with its own simplices. The connection of these codes to the polytopes II, is described in (1.5). We report a few formulas for ( N,( t, x) I and ] M,,(t) ] = C ] N,,(t, x) ( . Apart from the case n = 2,
FRANK UHLIG, TIN-YAU TAM, AND DAVID CARLSON
796
where one may list all codewords,
lM4k
- l)l=
they are based on computer
5578~”
&
+ 12,705~’
+22,520n4
JN2(u-l,x)l=u-
-18(u2
1)2 -
Difference
- 2)1=
+ 1008~’
--&u(1007u8
schemes show that
12(u2+
+ 1)(2x
if u - 1 ,< x < 2u - 2 [otherwise use (1.4),
- 1,2u
+ 19,614u6
+ 14,175
jr-u+ll,
INa(u - 1, x)1 = &[21(u2+
IA?+
counting of the N,( t, x):
1) + 4
- 3u + 3)” +5(2x
- 3u + 3)‘]
(1.3) and reduce to x = t = u -
+ 2766~~
+ 3759u4
+ 3844~~
11,
- 36).
1N4(t, x) 1 fits no polynomial formula, even for t < x <
3t.
PROBLEM.
Get formulas or other results for
1M,(t) 1 directly.
Part of the work was done while the authors were visiting at the University of Wisconsin - Madison with support from the research funds at the Norwegian School of Economics and Business Administration and the Norwegian Council for Science and Humanities (NAVF). The authors are grateful to these institutions, and to Hedge Tverberg fm essential references. REFERENCES 1
H. Anand, V. C. Dumir,
and H. Cupta,
2
(1966). W. H. Benson and 0. Jacoby, New Recreations with Magic Squares, Dover, 1976.
A combinatorial
distribution
problem,
Duke Math. J. 33:757-769
3
4
G. Birkhoff, Tres observaciones sobre el algebra lineal, Rev. Uniu. Nat. Tucuman Ser. A 5:147-151 (1946). FL A. Brualdi and P. M. Gibson, Convex polyhedra of doubly stochastic matrices, I. Applications of the permanent function, /. Combin. Theory A 22:194-230 (1977); II. The graph of Q,, J. Combin. Theory B 22:175-198 (1977); III. Affine and combinatorial properties of n,, J. Combin. Theory A 22:338-351 (1977).
AUBURN
1990 CONFERENCE
ON MATRIX THEORY
5
A. Brbndsted,
Continuous
barycenter
6
Math. 4:179-187 (1986). W. Dahmen and C. A. Micchelli,
functions
797
on convex pOlytOPes, Exposition.
The number of solutions to linear diophantine
equations and multivariate splines, Trans. Amer. Math. SOC.308509-532 7
E. Ehrhart, DBmonstration de la loi de &ciprocite Acud. Sci. Paris 265A:91-94 (1967).
8
B. Fuglede,
Continuous
Math. 4:163-178 9
D. M. Jackson
11
G. Knill, International I. G. Macdonald,
integer matrices,
SIAM].
standard book numbers,
Polynomials
Math. Sot. (2) 4:181-192
in the marketplace,
Amer.
Mbh.
associated
of generalized
double
Comput. 4:474-477 (1975). Math. Teacher 74:47-48 (1981).
with finite cell-complexes,
1.
London
(1971).
Combinatoq
Analysis I, ZZ, Cambridge
in one volume by bhelsea, 14
Modular arithmetic
and G. H. J. van Rees, The enumeration
nonnegative
P. MacMahon,
C. R.
(1988).
12 13
(1988).
selection in a convexity theorem of Minkowski, Exposition.
J. A. Gallian and S. Winters,
stochastic
rationnel,
(1986).
Monthly 95:548-551 10
du polyedre
U.P.,
1915,
1916; reprinted
1960.
K. Mano, On the formula of ,,H,,
Sci. Rep. Fuc. Lit. Sci. Hirosaki Unio. 8:58-60
(1961). 15
P. McMullen,
Valuations
and Euler-twe
relations
for certain
classes of convex
polytopes, 17
Proc. London Math. Sot. 35:113-135 (1977). N. J. A. Sloane, A Handbook oflnteger Sequences, Academic, New York, 1973. R. P. Stanley, Combinatorics and Commutative Algebra, Prog. Math. 41, Birkhlser,
18
E.
16
1983. F. Wood,
Self-checking (1987).
codes-an
application
of modular
Teacher 80:312-316
Received 19 August 1991; final manuscript accepted 30 August 1991
arithmetic,
Math.