Some extremum problems for matrix norms

SHORT CO:4MUNlCATIONS SOME EXTREMUM PROBLEMS FOR MATRIX NORMS* V. N.

FADEEVA

Leningrad 16

(Received

IN

[l .I,the

following

irreducible it

is

(using

required

problem

is

similarity

to

find

1966)

May

investigated:

given

transformations

a non-singular N (DAD-‘)

with

diagonal

a matrix

A that

a permutation

matrix

n such

iS

matrix).

that

inf N (DEAD,-I),

=

=a where no runs

through

the

Nz(C)=

class

of

all

non-singular

Sp CC’ = i

cijz,

matrices.

Here

for

matrix

c = (Ci,).

i,j=i

For positive norms

matrices

subordinate

In this

to

paper,

A, the

a Holder

we shall

problem

vector

consider

is

solved

norms

[31.

the

problem

in

of

finding

minIID-lADI/ D>O

for

the

norm II* II subordinate

matrix

(11 c (1 = max Jyx eigenvalue

of

the

Euclidean

where yi is a singular value matrix C’C), where we do not

the

The required

to

(1)

minimum does

not

always

exist,

norm of

the

vectors

of the matrix C, i.e. an assume that A is positive. since

linf I(D-*AD

I(

is

q>o always non-zero

’

Zh.

attained.

This

subdiagonal

v5;chisl.

is

true,

in fact,

for

triangular

elements.

(blot. mat.

Fiz.

7,

2, 401

203

- 404,

1967.

matrices

with

not

V.N. Fadeeva

204

Let

lij f 0. By means of a similarity transformation, ments may be made as small as we choose, so that as close less

as we choose

to the spectral

than the spectral

radius.

radius

Consequently

the non-diagonal ele11D-lAD )) may be made

max II iii i

II* but

cannot be

infll~-lADII = maxi Zif1, i

D>O

However, for

any matrix D

and &I2 +

-2

lal + - * * +L2 +

. . .

0

. . .

. . .

Let u be the greatest

eigenvalue

. . .

[

*

4

* * * +iT,22

0

. . . *

ha2

;i’xz

.. . .

0 -2 + Isa

. . .

.

1nn2 I

.., ..,

Let max

1Ziil = 1

j

Zjjl,

<

n.

of A'A.

i

Then 2 P 3 Ejj2+ f2-ki.j+

* * ++

tn j2

>

Ijj2,,

so that

I

For j = n. instead

Before considering second order matrix.

_.a

_

of A 'A it

is necessary

to form AA‘.

the problem in general, Thus, let

let us consider

it

for

a

Some

extremum

problems

We assume that b f 0 and c # 0, since while If b = 0, c f 0 (or vice versa), max(la(,

(d()

We note

is not

first is

matrix

norn~s

205

if b = c = 0, A is itself diagonal, A is triangular and inf IID-lADll = D>o

attained.

of all

that

of f 1, we have II El.@2 Eq-1A’AE2

for

similar

if El and E2 are diagonal

II = II A

II,

since

the matrix

matrices

composed

(E~AEz)‘(E~AEz)

=

to A’A.

Without loss of generality, we may assume that the non-diagonal elements of A are positive, since otherwise it is sufficient to multiply it by a suitable matrix El. Hence, let b > 0 and c > 0. We take D = [l,

ia].

Then D-1AD =

will be symmetric. But for a symmetric matrix, the norm under consideration equals its spectral radius and hence cannot be made any smaller. Thus, for a 2nd order matrix with non-zero super-diagonal elements, the diagonal matrix giving min 11D-lAD II is found in explicit form. D>O Let A be an n-th order matrix. Let us make the following assumptions. Firstly, that min D-~AD exists, and secondly, that the extremum

II

D>

II

o

matrix B = D1-lAD, simple. Let us consider

is such that the greatest

the family D-‘AD

where A = D1 -lD It

is evident

= [El,

...,

eigenvalue

~1 of B’B is

of matrices

= D-‘D,BDi-‘D

&,I is

that B is contained

= A-‘BA,

a diagonal

(2)

matrix.

in the family

(2)

for A = 1.

Let us denote the greatest singular value of the matrix A-lBA by Since ex hypothesi ~1 = ~(1, . . . . 1) is simple, the CI = c1(61, . . . . 6,). partial derivatives of u with respect to 61, . . . , 6a exist at the Point [I.

1, . . . .

11 and,

since

~1 is an extremum, they are zero at that point.

Let X be the normed eigenvector We now calculate these derivatives. of the matrix (A-lBA)‘(A-lBA), belonging to the eigenvalue CI. This means that

206

L’.N.

Fadeeva

A\B,A-‘BAX = pX, Differentiating

(3)

with

respect

?&-2BAx-

2AB’A-3

(X, X) = 1.

to 61,

we obtain

($1

1

while 8X

(ast > -,x

We now put Ii

Aj = I in

(4)

and v and X become

matrix

B’B belonging

rewrite

(4)

and

I,B’BX,

(5)

and (5).

=o.

with

- 2B’IiBXI

= CO, . . . , 1,

Then aA /‘a6i

1-11and X1, where to the

(5)

‘Y1 is the

eigenvalue

~1.

A = 1 in the

Let

. . . , fll =

eigenvector

of the

Ui = (%i/&;)jn=1;

we

form

+ B’BIi.7il _t B’BU, = “” I asi I&I

(6)

Xi + I*I”i,

while (Vi, Xi) = 0.

(‘i)

Taking the scalar product of (61 with the vector Xi and taking into account the fact that (.~,,,X ) = 1, (ui, x1) = 0, (B’BUi, K1) = (Ui, B’B.rl) = u(ui, y,) = 0, we obtain

But BX, = &Y1, where Y1 is the i3B’ belonging to the eigenvalpe g1.

We write equal to

eigenvector

Y1 = (xl, . . . . xu)‘. Y1 = (Y1. ..., Yb)‘. zero for i = I, 2. . . . , n, we obtain xi2

Let

normal Hence

Z== (!sI!~ . . . . fxclf’=

zzz

i=l

yp,

([Ill/,

.I.;

,

of the

Putting

matrix

(&QGi)

2, . . . , n.

\!/tzI)‘.

Then X, = E,Z,

Y,

= 6,Z,

1nzi

Some

where

El and E2 are

which

are

is

equal

one of the

to

extremum

two diagonal i- 1. Since

eigenvalues

for

problems

matrix

the

matrices, BE$ = j’&Z,

of the

matrix

207

norms

diagonal

elements

&BE,2 = y1Ez,

h = EZ8F1.

We also

of

i. e. iF*

have

the

in-

equality

which

follows

from the

fact

E1R’E2EZBE1 = E,-lB’8El values

hi

that

U’YI = Y$X,.

is similar

to the

Since

matrix

the

matrix

5’13, for

all

B’B = eigen-

of E we have

is the eigenvalue greatest in modulus of the matrix R~BEI, Hence G so that, on the assumption that ~1 Js simple, the least norm of a matrix of form D-lA;), n > 0 is equal to the spectral radius of E#El, which is similar to E2AE1. Then the matrix D1 which gives the extremum acquires the following significance. The matrices E2BEl and ElB’E2 have, vector Z with non-negative components.

as we have seen, a common eigenHence E2AEl and EI.+I‘E2 possess

eigenvectors

belonging to the eigenvalue Vul, /c the vectors n,z and n,-‘z also evidently, with non-negative components. Hence the respectively, diagonal elements of the matrix “I are determined as the square roots of the ratios of the corresponding components of these vectors. In conclusion, that Jv~ diagonal

we note

that

the

matrix

is an eigenvalue of one of the matrix with elements k I.

EZAEl is similar matrices

,AP, where

Trans

lated

to AElEz E is

so

a

by F.E. J. Rich

REFERENCES 1.

OSBORNE, E.E. ‘7, 4,

338

On pre-conditioning 1960.

of matrices.

J.

Ass.

comput.

Mach.

matrices

in

- 345,

2.

STOER, J. and WrR~GALL, C. Transformations by diagonal a normed space. Num. Math. 4, 2, 158 - 171, 1962.

3.

OSTROWSKI,

A.M. Gn matrix

norms,

,Math. z.,

63,

1, 2 - 18,

1955.