Reduction of a matrix using properties of the schur complement

LINEAR-

ALGEBRA

AND

ITS

23

APPLICATIONS

Reduction of a Matrix Using Properties of the Schur Complement* EMILIE

V. HAYNSWORTH

of Afathematics,

Department Auburn.

Auburn

University

Alabama

Communicated

by Alan

J. Hoffman

1. INTRODUCTION

Suppose A.

B is a nonsingular

The Schur complement

follows:

submatrix

of B in A, denoted

Let a be the matrix

tion of rows and columns leaving

principal

obtained

of an n x n matrix

by (A/B), is defined

from A by a simultaneous

which puts B into the upper left corner

the rows and columns

of B and G in increasing

as

permutaof A,

order.

Then (A/B) = G Schur proved that the determinant of any nonsingular

principal

DB-lC.

(1)

of A is the product of the determinants

submatrix

B with its Schur

complement,

IAl = IBII(W In

[l]

the following

“quotient

(2)

property”

was proved:

(A/B) = ((AIC)I(BIC)). * This work was done under contract DA-91-591-EUC-3686 of the U.S. Army with the Institute of Mathematics, University of Basel. The author wishes to thank Prof.

A. M. Ostrowski

for very helpful Linear

Copyright

0

discussions.

Algebra

1970 by American

and Its Applications Elsevier

Publishing

3(1970), Company,

23-29 Inc.

E. V.

24 In another

paper

[3] it was shown that

matrix can be determined together

the inertia

from that of any nonsingular

with that of its Schur complement.

and B is a nonsingular

principal

of a Hermitian

principal submatrix

That

submatrix

HAYNSWORTH

is, if A is Hermitian

of A, then

In A = In B + In(A/B).

(3)

In this paper we make use of the properties to compute

the eigenvalues

n, from those 2. APPLICATIONS

of related OF

THE

It is well known a matrix

of an arbitrary

matrices SCHUR

[2] that,

of the Schur complement

complex

matrix

of order m and n -

COMPLEMENT

TO

EIGENVALUE

if there is a commutator

A of order n to a matrix

A, of order

m, respectively. PROBLEMS

Y of rank m from

B of order llz,

AY=YB, then m, of the roots of A are the roots of B. we can obtain

a matrix

A by first constructing replaced

by those

the matrix Since

whose roots a matrix

In Theorem

are the remaining

A(Y)

in which

of Y, and then computing

Y, in A(Y).

(A(Y)

is defined

Y has rank m, we assume,

Y= where Y, is a nonsingular

m

x

the Schur

loss of generality,

For, if it is necessary

matrix

(PAP-l)(PY)

Linear

to permute

in the first m rows, we may since the equation

= (PY)B

(Fi), we partition

Y,

where A,,

that

to (4).

Now under the assumption with

of

(4

permute the rows and columns of A in the same manner,

is equivalent

complement

9

m matrix.

of

of A are

below.)

(~,Yl1

the rows of Y to obtain a nonsingular

m roots

m columns

precisely

without

1 we show that 12 -

is an m x 112 matrix.

Algebra

and Its

Applications

3(1970),

23-29

the matrix

A conformally

PROPERTIES

OF THE

We define

THEOREM

A(Y)

1.

m, respectively.

SCHUR

25

COMPLEMENT

to be the n x wz matrix

Suppose

A and B are square matrices of orders rz and

If there exists an n x m matrix

Y, of rank m, satisfying

(4) and (5), then m roots of A aye roots of B and the remaining n aye those of (A(Y)/Y,),

the Schw

complement

m roots

of Y, in the ma&ix

A(Y)

defined in (6). Proof.

From

(4) we have

A,,Y, + A,sY, = Y,B> (7)

A,,Y, + A,$‘, = Y,B. Now we let

Yl 0 y, I?&., i

1

s= Since

Y,

is nonsingular,

S is nonsingular. AS =(;:;

Then,

from

(7),

;;:).

If we let

then Y,B

sa= so that

S-lA.S

Y,B

Y,C

’

= a if and only if Y,C = A,,

which implies

Y,C D+

and

D + Y,C = A,,

that D = A,, Linear

Y,Y,-lA12,

Algebra

and Its Applications

3(1970),

23-29

26

E. V. HAYNSWORTH

the Schur complement

of Y, in the matrix

A(Y)

in (6).

Since the roots

of a are those of A, which are in turn those of the submatrices the theorem

B and D,

is proved.

COROLLARY. Suppose the matrix A has m linearly independent column to the roots, 2,, . . . , 1, (which are not

characteristic vectors corresponding necessarily

distinct).

Let X be the n x m matrix whose columns are these

vectors and assume, without loss of generality,

where X, is nonsingular.

Then the remaining

of the matrix

where A(X)

Proof.

(A(X)/X,),

Since

AX

= XA,

follows immediately

that

n -

where A = diag(il,, . . ., A,),

Often it is easier to use some linear combination

this reason

LEMMA

the corollary

from the theorem.

vectors corresponding in order to compute

m roots of A are those

is constructed as in the theorem.

of the characteristic

to certain roots, rather than the vectors themselves, the matrix

we prove

which contains

Lemma

1. If A E M,(C),

the remaining

roots.

For

1. BE M,(C),

X is a commutator of rank m

from A to B, and Y is an31matrix whose columns are also a basis for the space generated by the columns of X, then Y is a commutator from A to a matrix

2 which is similar to B.

Proof.

Y is a basis for the column

space of X if and only if

Y=XC, where

C is a nonsingular

matrix.

AY = AXC

Then

= XBC

= XC8

=

yB,

where E? = C-lBC. In particular,

if A has its elements

in a given field F, but the roots

of A do not lie in this fieId, then in certain cases, by using a linear combinaLinear Algebra and Its Applications

3(1970), 23-29

PROPERTIES

OF

THE

tion of the vectors, done entirely in F. of degree

2

SCHUR

COMPLEMENT

the computation

27

of the Schur complement

can be

The case in which the roots lie in an extension

over F is proved in Theorem

we need also the result of Lemma

2.

field

For the proof of Theorem

2, which is undoubtedly

2,

well known,

but as its proof is so simple it is included here for the sake of completeness. 2. Su$$ose A is an n x n matrix over a field F having as

LEMMA

a characteristic root il = a + bvd, zwheye a, b, and d belong to F but VZdoes not.

Then (i) the characteristic vector corresponding

X + VdY,

where X and Y aye linearly

(ii) x = a -

to il can be written as

independent

vectors ove7 F, and

b d is a root of A which has the corresponding

vector, X -

characteristic

Vii.

Proof.

Statement

(i) follows

R lies in the extension

field ~(j/d),

must also lie in this field.

A(X + which implies

immediately

from the fact that, since

th e corresponding

characteristic

vector

Then

@Y) =

(a + bva)(X

+ VZY),

that AX=aX+bdY

AY=bX+aY.

and

Thus A(X which

-

VZy) = aX + bdY -

proves

statement

vd(bX

+ aY) = (a -

(ii).

2. Suppose A is an 1z x n matrix

THEOREM

the distinct roots & = ai + b,v& but jldi does not belong to F. any pair i, j.

bVd)(X -VY),

over a field F having

i = 1, . . . , k, where ai, bi, d, belong to F,

Suppose also that iii # xj = aj -

b,Vd, for

Then.

(i) A has the 2k distinct roots, ai -+ b,Vdi, with corresponding Xifvd;Yi,i=l,..., independent (ii)

vectors

k, where the 2k vectors Xi, Yi belong to F and a7e over F.

If we let Z be the n x 2k matrix having the vectors Xi, Yi as its

coluvnns, the other n -

2k roots of A can be found as roots of the Schur Linear

Algebra

and Its Applications

3(1970), 23-29

E. V. HAYNSWORTH

28 complement of the matrix 2, in A(Z),

where A(Z)

is constructed as in the

theorem. Statement

Proo/.

roots

are distinct.

(i) follows immediately Then,

from Lemma

2 since the

if we let Wi be the n x 2 matrix,

wi =

viiiYi, xi

(Xi +

- vi&Yi)

and let W be the n x 2k matrix, W = (W,, w,, . . . >W,),

we have

4 W = WA,

where

A = diag(ii,,

I,, . . ., A,, I,).

Now we let

Ci be the 2 x 2 matrix

Then

WiCi = 2Z,, where Zi = (Xi, YJ.

Thus,

if we let C =

~~=I

* Ci,

having

the

we have WC = 22. and, by Lemma 2k roots of A. Theorem of 2,

1, Z is a commutator Then, using the columns

1 will have its elements

in A(Z)

from A to a matrix of Z, the matrix

in F, and thus the Schur complement

will also have its elements

This application field of rationals,

A(Z) defined in

is particularly

useful,

in F. of course,

where Vii is not rational,

if the field F is the

or the field of reals, where

v-.d, IS not real. Example.

Consider

the matrix

which has the roots 5 and (5 + roots Linear

we have the vectors Algebra

and

Its

1/5)/2. Corresponding

X 4

Apfhations

1/5 Y, where 3(1970), 23-29

to the two irrational

PROPERTIES

OF THE

SCHUR

29

COMPLEMENT

Then

and (/4(2)/Z,)

0 1) i 4

= 4 + $(l

Iii

1 1

1 o = 5.

REFERENCES 1 D. Crabtree and Emilie Haynsworth, a matrix,

Proc.

Amev.

Math.

Sot.

An identity

for the Schur complement

of

(to appear).

2 L. S. Goddard and H. Schneider, Pairs of matrices with a non-zero commutator, Proc.

Phil. SIX. 61(1955), 551-553. Haynsworth, Determination of the inertia

Camb.

3 Emilie

matrix, Linear Received

May

Algebra l(1968). 9,

of a partitioned

Hermitian

73-81.

7969

Linear

Algebra

and

Its

Applications

3(1970), 23-29