On a characterization of tridiagonal matrices by M. Fiedler

LINEAR

On

ALGEBRA

AND

ITS

a Characterization

WERNER

APPLICATIONS

of Tridiagonal

C. RHEINBOLDT

AND

Computer Science Center

8, 87-90

Matrices by M. Fiedler*

ROGER A. SHEPHERD Department of Mathematics

University of Maryland

University of Maryland

College Park. Maryland

College Park, Maryland

Communicated

87

(1974)

by Hans Schneider

ABSTRACT In this note two new proofs are given of the following characterization M. Fiedler:

Let C,, n > 2, be the class of all symmetric,

with the property that rank (A + D) > n for any A E C, there exists a permutation and irreducible.

Recently,

theorem of

real matrices A of order n

1 for any diagonal real matrix D. Then matrix P such that PAPT

is tridiagonal

M. Fiedler [l] obtained the following characterization

of tridiagonal

of a class

matrices.

THEOREM.

Let C,, n > 2, be the class of all symmetric, real matrices A

of order n z&h the property

that rank(A

+ D) 3 n -

1 for any diagonal

real matrix D. Then for any A E C, there exists a permutation matrix P such that PAPT

is tridiagonal and irreducible.

The purpose of this note is to give two new proofs of this result which appear to be more direct than the proof by Fiedler. A preliminary must

and self-evident

be irreducible.

symmetric submatrix

Gaussian in C,.

The

remark

observation

elimination

More precisely,

applied

is that

central

any matrix

to both

to a matrix

A in C,

proofs

in Cnfl

is that

leads to a

for any index k, 1 < k < n + 1, we can

choose d, such that ulclc+ d, # 0, so this diagonal element

can be used as

*This research was supported in part by Grant GJ-106’7 from the National Science Foundation to the Computer Science Center of the University of Maryland. 0 American

Elsevier Publishing

Company,

Inc., 1974

88

WERNER

C. RHEINBOLDT

AND ROGER

A. SHEPHERD

the pivot for an elimination step. Since simultaneous permutation of the rows and columns does not lead out of Cnfl, it is no restriction to assume that k = 1 and that A E C,+l is partitioned as follows:

(1) all let

Here the irreducibility of A ensures that a # 0. Now with dr # a=-

1

(2)

all + dl ’ where I is then x n identity matrix. Then for any D = diag(d,, ds, we obtain L(A + D)LT = with

b =

all + dl o

diag(d,,. . ., d,+l).

.?4Lll)

B = A - aaaT,

(3)

From

n < rank(A + D) = rank L(A + D)LT = rank(B +

b) +

1,

it now follows that B E C,. This opens up the possibility of proving the theorem by induction on n. For n = 2 the statement is trivial, and it is useful to note that it also holds for n = 3. In fact, in that case the irreducibility requires that aij = aji = 0 for at most one index pair i # i. For, if aij # 0, i, i = 1, 2, 3, i # j, then for

4 =

-all

+-,

a21al3

a23

4 =

-az2

+-,

42923

d, = -a33

+F,

a13

the matrix A + diag(dl, d2, d3) has rank one and hence does not belong to cs. For Suppose that the statement holds for n > 3 and let A E C,,,. the first proof we assume that by simultaneous row and column permutations A has been brought into the form

aoT

=

b2,...,

al,+d,

alj #

0,

j =

2,...,m+

1,

(4)

TRIDIAGONAL

where

MATRICES

89

no row of A has less than m nonzero

the square matrices For any dr # -

off-diagonal

aI1 the matrix

^

of B,,

may assume that dr # nonzero. matrix

Bll

= A,,

elements.

B is an irreducible,

By irreducibility,

are at least m nonzero

off-diagonal

This implies that m < 2. at least m nonzero,

m = 2, for otherwise

A;

elements step;

off-diagonal

in one of the first m rows m 3 n -

n

x

n

with an edge between

matrix vertex

elimination

(3) is obtained

A, that

is, the graph j exactly

[2] on changes

step.

In terms

matrix. structure

G(A) on n if i # j and

in the structure

of our elimination

graph G(B) of the matrix

from G(A) by deleting

vertex

1 together

B

with all

with it and by adding an edge between any vertices i and i,

i # i, which were both adjacent The characterization path.

the case

irreducible

of the (undirected)

i and vertex

We use a result of Parter

due to a Gaussian

edges incident

This excludes

tridiagonal,

step (3), this result states that the structure in Eq.

2 > 0 rows of A

by choice of m, they

at best one row of B has less than two such elements.

of a symmetric

aij = aj, # 0.

2 rows with two, nonzero

elements.

The second proof is based on the notion vertices

of B1, is

permuted tridiagonal

therefore,

Thus, m = 1 and A is indeed a permuted graph

T.

is not zero, and hence there

The last n -

were not changed in the elimination contain

ccaOa,,

can be zero only for a single value of dr, we

and thus has two rows with one, and n -

of B.

graph

-

all was chosen so that every element

By induction hypothesis,

off-diagonal

and

m, respectively.

B in Eq. (3) then has the form

,

Since an element

elements,

and AZ2 have order m and n -

A,,

theorem

to vertex states

We proceed again by induction

1 in G(A).

that for A E C,, G(A) must be a

starting

from the earlier observation

that the result holds for n = 2, 3. Moreover, we use once more that for any A E Gn+r a symmetric produces a submatrix

Gaussian

elimination

B for which-by

step with any diagonal

induction

hypothesis-G(B)

pivot is a

path. Observe degree

first that

greater

than

for A E Cn+l, 12 > 3, no vertex two.

vertices i, j, 1 were adjacent

In fact,

some d, # -

a ,,-contains

hence is not a path. Since the irreducibility

that

in G(A)

three

to the vertex k. Then the elimination

aRk + d, # 0 as pivot produces -for

suppose

of G(A) can have

a resulting

structure

a three-cycle

Thus, all vertices

graph

distinct step with

G(B) which

with vertices

i, j, 1, and

of G(A) have degree at most two.

of A is equivalent

to the connectedness

of G(A),

WERNER

90

C. RHEINBOLDT

AND ROGER

this implies that G(A) is either a path or an (n + 1) cycle. case an elimination structure

step on any vertex would produce

graph G(B) which contradicts

B E C,.

A. SHEPHERD

In the latter

an n-cycle

as the

Thus, G(A) must be a

path and the proof is once more complete. REFERENCES 1 M. Fiedler, A characterization

of tridiagonal

matrices, Lin. Alg. and A@Z. 2(1969),

191-197. 2 S. Parter, Use of linear graphs in Gauss elimination, Received November,

197 1

SIAM

Rev. 3(1961), 119-130.