Inertia theory for simultaneously triangulable complex matrices

LINEAR

ALGEBRA

Inertia Theory Matrices*

AND

RICHARD

D.

University,

131

APPLICATIOXS

for Simultaneously

Idaho

State

ITS

Triangulable

Complex

HILL

Communicated

Pocatello,

by Alan

Idaho

J. Hoffman

INTRODUCTION

I.

The inertia

of a complex

matrix

A is defined to be the ordered triple

In A = (zc, v, 6) where z is the number real part,

v the number

zero real part.

with negative

of eigenvalues real part,

of A with positive

and 6 the number

with

Two classical results in inertia theory are those of Sylvester

and Lyapunov. SYLVESTER’S

If P is nonsingular and H is hermitian, then

THEOREM.

In H = In PHP*. LYAPUNOV’S

If A is a complex

THEOREM.

an H > 0 (positizje definite)

These results were generalized [8] to what MAIN

is called

INERTIA

1.

THEOREM.

2.

eigenvalues) .

With H > 0, Schneider Lyapunov *

theorem

Dedicated

by Ostrowski-Schneider

[5] and Taussky

Given a complex matrix A there exists

[6, p. 131 has generalized of Stein

1969

Algebra

(among others)

[7] to the following

A. M. Ostrowski

Linear 0

6(A) = 0 (i.e., A has no pure

If AH + HA* > 0, then In A = In H.

and that

to Professor

Copyright

then there exists

the

a hermitian H such that AH + HA* > 0 iff imaginary

matrix,

such that AH + HA* > 0 iff In A = (n, 0, 0).

and

by American

on his 75th

birthday.

Its Applications Elsevier

the

theorem.

2(1969),

Publishing

131-

Company,

142 Inc.

R. D. HILL

132 SCHNEIDER’S THEOREM. order

A, C, under

a natural

1, 2, . . . ) s.

Then

T(H)

Let A, C,, C,, . , ., C, be complex matrices

can be sim&aneously

n which

correspondence

there exists

= AHA*

-

triangulated.

i

be GQ,yi(k), i = 1, 2, . . . , n and

alz H > 0 such

C,HC,*

> 0

of

Let the eigenvalues

of k =

that

iff

k=l

(i = 1,2, . . . , n) Schneider

shows that

there

exists

no analogous

result

for his

T(H)

with H hermitian. As in

[l]

and

DEFINITION.

[2], we define

The

complex

quasi-commutativity

matrices

as follows.

A,, A,, . . . , A,

iff each of A,, A,, . . . , A, commutes

quasi-commutative

are

said to be

with AiAj

-

A,Ai

(i, j = 1, 2,. . .) s). In this where

paper

we develop

A,, . . . , A,

In Theorems that

c;,i=l

are

be positive

d,,AiHAj*

Schneider-Taussky

condition

theorem

Our results in Section Schneider-Taussky

of negative

of values

in

general A,.

values,

Linear

Algebra

d,,A,HAj*

under

We conclude

A,,..., quasi-commutativity the individual

These

a natural

of positive

and 6 the number

;

values,

of zero values

of

If n(D) > 1 or Y(D) > 1, we show

matrices

A,, A,, . . . , A, and a hermitian

> 0 and In H is not equal to this ordered

dzi&(i)jdj)

(k = 1, 2, . . . , n). to show that

a simultaneous by discussing

can be relaxed

Its Applications

correspondence

with n(D) < 1 and y(D) < 1;

these results

triangulability conditions

to simultaneous

theorems. and

generalizations

the second part of the Ostrowski-

IV we give an example

hold

conditions

If A,, A,, . . . , A, are quasi-commutative

(k = 1, 2, . . . , n).

of c&l

is hermitian.

with H > 0 and then

In H = (n, v, 6) where 7~ is the number

cf,j=l

In Section

first

d,,A,HAj*

respectively.

III generalize

that there exist quasi-commutative triple

D = (dij)

and sufficient

of H hermitian.

22’). nk(‘), . . . , AkcS)under

1 d$ik(‘)&(j)

H such that

definite,

> 0; and D is hermitian

dijA,HA,*

Y the number cf,j=

and

for cE,j=l

theorem and the first part of the Ostrowski-

theorem.

eigenvalues

we show that

theory

quasi-commutative

take the forms of the Lyapunov

~~,i=l

inertial

3 and 4 we give necessary

with the much weaker

with

the

2(1969). 131-142

do not

hypothesis

on D under

on

which

triangulability

in

INERTIA II.

133

THEORY

THE CHARACTERIZATION

THEOREMS

In this section we shall prove two propositions for our two theorems. the theorems. theorems

Moreover,

of Section

direct

combine

propositions

than do

are used as lemmas

for the

1-5 we let A,, A,,

matrices

whose

. . ., A, be given as quasi-

eigenvalues

under

a

natural

are AA(‘),Ah(‘), . . , Ah(‘) (k = 1, 2, . . . , n) and let D = (cl,?) of order

A represent

represents

which naturally

reveal more structure

III.

complex

correspondence be hermitian

propositions

both

For results

Convention. commutative

Let

These

s. the

the Kronecker

computation

n x sn matrix product

(A,, A,, . . ., A,).

of the matrices

If DxH

D and H, we verify by

that

2

= A(D x H)A*.

d,A,HAj*

E,j=l We

adopt

the

following

notation

for the

remainder

of this

paper.

Let

2

QDxH(A) =

dijA,HAj*

i,j=l and

In {A, D} is defined

DEFINITION.

to be the

ordered triple (z, v, 6)

where 7z is the number of positive values, v the number of negative and 6 the number

of zero values

Since D is hermitian, to its conjugate

values,

of QD(&) (k = 1, 2, ...,n).

a simple calculation

shows that Q,(A,)

is equal

and is thus real (k = 1, 2, . . . , n). Thus, the above defini-

tion is meaningful. PROPOSITION (k =

1, 2,

PROPOSITION (k=1,2

1.1.

If H is hermitian

and QDxH(A) > 0, thenQ,(A,)

# 0

* . . , 75).

)...)

1.2.

If

H > 0

and

QDxH(A) > 0, then QD(jlk) > 0

n). Linear

Algebra

and

Its

Afiplications

2(1969), 131-142

134

R. D. HILL

PROPOSITION 2.1. If Q&,) # 0 (R = 1, 2, . . ., s), then there exists a hermitian H such that QDxH(A) > 0 and In H = In {A, D}. PROPOSITION 2.2. If QD(A,) > 0 (k = 1, 2,. . ., n), then there existsan H > 0 such that QDxH(A) > 0.

We remark that both parts of Proposition 2 are proven with the weaker condition of simultaneous triangulability on A,, A,, . . . , A,. THEOREM3. Q&)

#O

(k=

THEOREM4. iff Q&l,)

There exists a hermitian H such that QDxH(A) > 0 iff 1,%...,4.

There exists a positive definite H such that QDXH(A) > 0 1, 2, . . ., n).

> 0 (k =

Proof of 1.1. Our proof is by contraposition. We assume that A,, A,, . . . , A, have eigenvalues A,(‘), ;2k(‘),. . . , Ak(s) (under the given natural correspondence) such that QD(jlk)= 0 for some fixed k. Now by [2, p. 2251 there exists a (row) eigenvector v such that vAj = jlk% (i = 1, 2, . . . , s). Thus

v[QmAA)Iv* = iIQ&)lvHv* = 0. This is a contradiction to our hypothesis that QUxH(A) > 0.

n

Proof of 1.2. Letting Ah(‘),&@I, . . . , ilk(‘) (k = 1, 2, . . . , n) be a natural correspondence of eigenvalues of A,, A,, . . . , A,, we again apply [2, p. 2251. Then for each k there exists a (row) eigenvector vksuch that vk[QDx H(A)]~,* = [QD(&)]vkHvk* (k = 1,2, . . ., TZ). Since H > 0 and QIjxH(A) > 0, the are positive (k = 1, 2, . . . , n). Thus, numbers v,Hv,* and %[QDxH(A)lvk* the numbers QD(Ak) (k = 1, 2, . . . , B) must be positive. H

For our proof of 2.1 and 2.2 we need the following LEMMA. If A,, A,, . . ., A,

are sim&aneously

triangulable

complex

matrices whose eigenvalues under a natural correspondence are jlkC1),jlkC2),. . . ,

Akcs)(k=

1,2,...,

that SA,S-l

= A, +

/@I < E and Aj = Linear

Algebra

n), then given E > 0 there exists a nonsingular Tj where Tj = (t$)) is strictly u@er

diag {A1cil,. . . , A))}

and Its Applications

(i, I=

Z(1969). 131-142

1,2,.

S such

triangular with

. . , n; j = 1,2,. . . , s).

INERTIA

THEORY

135

triangulable, Since A,, Aa, . . ., A, are simultaneously exists a nonsingular Q such that QA#-’ = rlj + Bj where Bj = Let R, = diag (1, d-l, P, (j = 1,2,. . .) s) is strictly upper triangular. ~1’~“> where 6 > 0. By computation we have that R,(QAjQ-l)R,-l = T, where T. = (t$ satisfies t.$) = 61-ib$) if I> i, = 0 if 1 < i. Proof.

there (b$‘) . . ., Aj +

Now,

given E > 0,‘we choose 6 such that 0 < 6 < min {E/M, I> where M = max jbij)j (i, 1 = 1, 2, . . . , pz; i = 1, 2,. . ., s). Then \tf{‘j = \S’-“I$‘l < F Thus, S = RdQ is our required nonsingular matrix. Proof of 2.1. Since A,, A,, . . , A, are given to be simultaneously triangulable, by the above lemma given E > 0 there exists a nonsingular S such that S-iA,S = Ai + T, (i = 1,2, . . . , s) where /Ij =diag{ Al(j), jla(j), . . . , ;I ?% (j)} and Tj = (@) is strictly upper triangular with It$)I < F. Let. So = sgn [Qo(&)] (k =l, 2, . . , a). Since D is hermitian, we have seen 12). Let fi = diag {sr, ss, . . . , sn}. Then that Qo(&) is real (k = 1,2,..., A is hermitian and InE?=In{A,D). Let A, = S-lA,S c

where

Aj* = S*Ai*S-l*

= ilj + Tj. Then

d,,A,AA,*

=

i,j=l

i

(I)

d&l;

+ Ti)l?(ij

= /ij + T,* and

+ Tj*)

i,j=l

of E can be made

all elements

arbitrarily

= diagN?&d/~lQ&,)l,..

>o

since

small.

Now

.>lQ,(J,)l)

C?,(k) f 0

(k=1,2

,...)

7%).

Since T’! Lz,j=l dijA,lhij is fixed, for sufficiently small E > 0 we have that C;,,_l diiAiAAj* > 0. Thus, C%,j-;l dij(S-lAiS)A(S*A,*S-l*) > 0. Applying Sylvester’s theorem, s

S

i&

dij(S-'AiS)I?(S*A

[, Linear

j*S-l*)

1

S* > 0

Algebra and Its A+plications

2(1969),

131-142

136

K. D. HILL

or i

dijA,HAj*

where

> 0

H = SI?S*.

i.j=l

Again

appealing

to the Sylvester InH=

Combining

InA.

(2)

(1) and (2) gives us that InH=

Proposition

In{A,D}.

2.2 is a corollary

us that In {A, D> = (n, 0, 0), THE SECOND-PART

III.

theorem,

The inertias

n

to 2.1 since the hypothesis

which,

OSTROWSKI-SCHNEIDER-TAUSSKY

of the matrix D = (dij) of c&i

which of these polynomials

in A,, . . . , A,, A,*,

GENERALIZATION

d,,A,HAj*

InH=In{A,D}

result as stated in the following theorem. of Section

5.

IfdD)

<

determine

. . ., AS*, H possesses the

5 the convention THEOREM

of 2.2 gives

by 2.1, must be the inertia of H.

(For Theorem

II is still in effect.)

1, v(D)

<

1, and

QDxH(A) > 0, then In H =

In (A, D}. No less restrictive result.

hypothesis

on D will give the In H = In {A, D}

This is the content of the following theorem.

Note that commuting

in pairs is a stronger condition than quasi-commutativity THEOREM

complex

6.

matrices

a hermitian

If n(D) > 1 OY v(D) > 1, and A,, A,, . . . , A, of order

n which

on A,, A,, . . . , A,.

n > 2, the% there commute

H of order n such that QDxH(A) > 0 and

in pairs

exist and

In H # In {A, D}.

That H be nonsingular is not only a necessary condition that Theorem 5 hold, it is also used in the author’s

proof

For hermitian K it is known that n(AKA*)

of this result. < n(K).

Since this result

does not seem to appear in the literature we include a proof of the special case which we shall use in the following

lemma, viz. where A is n by sn

and K is of order sn. Linear

Algebra

and Its Applications

2(1969),

131-

142

INERTIA

If we denote singular Then

the rank of A by r, then r < n.

P and Q such that AKA*

have

137

THEORY

that

consisting

= P-l(E,,

n(AKA*)

0,. . ., O)Q-lKQ-l*(E,,

= n(L,,

equalities.

submatrix

Thus, If

LEMMA.

where

L,,

and

we

is the matrix

of L = Q-lKQ-l*.

of L, zz(L,,)
n(AKA*)

there exist non-

0,. . ., O)TP-l*

@ Ox_,) = n(L,,)

of the first Y rows and Y columns

is a principal

Thus,

PAQ = (E,, 0, . . ., 0) where E, = II @ OS_,.

Since L,,

by one of the Cauchy

in-

< n(L).

QDxH(A) > 0 where D = (dij) is hernzitian, n(D) <

I,

and v(D) < 1, then H is nonsing&ar. Proof.

We

have

observed

both D and H are hermitian, are real.

Thus,

using

that

QDxll(A)

x H)A*

n
=

We

note

which

> 0 by hypothesis,

= n.

n(H)

if

n(D) = 1

and

v(D) = 0

I v(H)

if

z(D)

and

v(D) = 1.

that

n, n(H) + v(H) < n.

H is nonsingular. out that

hypothesis

they

= 0

if n(D) = v(D) = 0, our lemma

We wish to point

Hence,

is vacuously Thus,

satisfied.

n(H) + v(H) = n

n

no quasi-commutativity

on A,, A,, . . . , A, is needed

or simultaneous for the lemma.

need not even be square.

Proof of Theorem 5.

Thus,

x H)A*]

+ v(D)y(H)

H is of order

Q,(&)

n[A(D

n(D) = v(D) = 1

triangulability

that

+ v(D)v(H).

if

implies

In fact,

Since

n(H) + v(H)

that

Since

x H)A*.

and thus its eigenvalues

[4, p. 241,

n(D x H) =s-c(D)n(H) Since A(D

= A(D

D x H is hermitian

Since QDxH(A) > 0, by Proposition

1.1 we have

# 0 (k = 1, 2,. . ., n). by Proposition

2.1 there

exists

a hermitian

Linear Algebra and Its Applications

H,, such

that

2(1969), 131-142

138

R. D. HILL

and InHa= H as H,.

We rename

In{A,D}.

(4

Then

2

diiAiHIAj*

> 0.

(5)

I,j=l

H, = tH, + (1 - t)H, where t E CO,11.

Let

i

d,,AiHtAj*

= t i

t,j=1

which

dijAiHIA,*

Then

+ (1 ~ t) 2

is positive

definite

By the lemma,

d,,A,H,AI*

z,j=l

i,j=l

by (3) and (5).

H, is nonsingular. Since H, varies continuously with of H, vary continuously with t. Thus, H nonsingular

t, the eigenvalues for t E [0, l] implies

that In H, = In H,.

Since

H, = H, combining

(4) and (6) we have InH

Proof of Theorem 6.

(6)

= In{A,D}.

that

n

Since D is hermitian, there UDU* = diag {,I_+,,us, . . . , ,u,,} where ,~r andpz are the two positive eigenvalues of D guaranteed by the hypothesis that n(D) > 1 and ,~a,. . . , pu, are the other eigenvalues of D. We define the matrices A,, A,, . . . , A, and H by exists

a unitary

Suppose n(D) > 1.

U = (u,,) such that

A, = uljI $- ZS~~E,,~ where I is the identity in its (n, 1) position

matrix

(j = 1,2, . . .) s)

and E,, is the n-square

and zeros elsewhere H=diag{l,l,...,

and l,O}.

Then

= diag

2

~r~d~~ti,~,. .

. ,

i,j=l

Linear

Algebva

and Its Applications

2(1969),

131-142

matrix

with a one

INERTIA

139

THEORY

= diag {p,, . . .,/-Q,pz)> 0 since ( UDU*)kl =

c:,j=l

uUkzdiitilj= ,u& 6,, = 0

where

if

k#l

if

k = 1.

=l

Thus, QDxH(A) > 0 with H singular. The case v(D) > 1 follows taking

H instead

-

Since H is singular, that Q,(A,) Thus

In H f

EXTENSIONS

6(H) > 0.

However,

Thus,

A,, A,, . . . , A, commute

SIMULTANEOUS

1.1 is basic to the proofs

to simultaneous

Example.

Let A1=(:

dijA,HAj* -

A,A,

commutative.

They

The conclusions triangulability conditions

of Theorem

3 and 5 whereas

not

hypothesis

example

on these results cannot

triangulability.

b),A2=ti

> 0 with ztj=i

does

+

n

TRIANGULABILITY

shows that the quasi-commutativity

c&=,

in pairs.

1.2 is basic to the proof of Theorem 4. The following

be relaxed

by

1.1 gives us

Proposition

we have that A,A, and AjAi are both equal to ulpliI

TO

Proposition

A,A,

argument

{A, D}.

(z+u,~ + u.~~u~JE,,.

Proposition

from the above

0 (k = 1, 2, . . ., n), i.e., if In (A, D} = (n, Y, a), then 6 = 0.

f

Ry computation

IV.

immediately

of H.

commute

dijl,(‘)J,(j) with

are obviously

and H=D=I,.

Then

= 0. Since the commutator

A,, A,

and A,

simultaneously

of Propositions

if certain conditions

i)

are not quasi-

triangulable.

1.1 and 1.2 hold under simultaneous on the inertia of D are added.

These

differ for 1.1 and 1.2.

We use the following

theorem which was communicated

to the author

by Carlson : THEOREM.

Let A,, A,, . . . , A, be simultaneo&y

triangulable complex

matrices of order n whose eigenualues wader a natural correspondence are p,

Ad”‘, . . . ) 22’) (k = 1, 2, . . ., n); Linear

Algebra

let H be hermitian of order n; and

Its

,4$$lications

2(1969),

and

131-142

R. D. HILL

140 let D be hermitian

of order s with eigenvalues

exist simultaneously

6,, 6,, . . . , 6,.

Then there

triamgulable matrices B,, B,, . . . , B, of order n with

eigenvaluespu,(‘), auk, . . . , pfi(‘) (k = 1, 2, . . . , n) under a natural correspondence such that

kld,lAiHAj*

=

g16iBiH~i*

and i

dijjlk(i)Xj) =

r,j=

Proof.

1

2

BI!,uk(i)12

(k = 1,2,

. . ., n).

I==1

Let D = UAU*

where A = diag {a,, . . . , S,}.

. . ., AJD x H(A,,

(4 =(A,,...,

A,)@’

x l)(A

=(B,,...,

BJA

x H(B,,

Then

. . ., A,)* x H)(U

x I)*(A,,

. . ., A,)*

. . ., B,)*

triangulable where B, = cjzI uijAi. Then B,, . . . , B, are simultaneously with eigenvalues ,uk(j) = zi=, uUij;ikCE) (k = 1, 2, . . . , n). A short computation with the ,~k(~)completes the proof. n If YE(D) = 1 We first consider the Proposition 1.2 generalization. and Y(D) 3 1, we use the above theorem to get the “only if” part of the Schneider theorem (as stated on p. 132). Wielandt [7, p. 161 has given a short matrix-theoretic proof of this result. Extensions of the above counterexample rule out any simultaneous triangulability generalizations withn(D) > 1. We summarize in the following table the conditions on In D which with H > 0 and QDrH(A) > 0 imply

QD(l,) > 0 (k = 1, 2,. . ., n).

that

TABLE

1

x

1

X

s.t.

q.c.

X

s.t.

9.c.

31 x

I s.t.

0

denotes

that

QD~H(A)

q.c.

> 0 is

impossible. Linear

Algebra

and Its Applications

2(1969),

131-

142

INERTIA

141

THEORY

Again using the Carlson theorem and extensions example,

we generalize

of the above counter-

1.1. We summarize

Proposition

table the conditions

on In D such that H hermitian

imply

0 (k = 1, 2,. . ., n).

that QJL,)

TABLE

2

..,nm

.

f

’

‘\

Y(D) 0

I

(’

I

~

X

s.t.

,

>l q.c.

I

1

s.t. ~ ct. ~ q.c. I q.c. q.c. ~ q.c.

>l

Following

through

see that Theorems Theorem

the proofs

as previously

1.

Remarks.

given in this paper,

3 and 5 hold under the conditions

4 holds under the conditions

hypothesis, Q.

in the following

and QDxH(A) > 0

Any

time

that

of Table

we

use

we

of Table 2 whereas

1.

the

we could use the slightly weaker condition

quasi-commutativity of Drazin’s property

See [l]. If we let D =

2. becomes

Lyapunov’s

, A, = I, and A, = A, then Theorem

theorem

whereas

Theorems

two parts of the O-S-T Main Inertia Theorem.

A, = I, and A, = A, then Theorem Theorems

D=

3 and 5 become

1

0

i0

-1s’

3.

we get Schneider’s

1

are all l’s

A,HAj*

If we let D =

has one positive

In H = (n, 0, 0) by Theorem

i0

0 -1’ i [7] and

theorem.

If

to Theorem

5:

T(H).

the following

> 0, then H > 0.

the

1

Stein’s theorem

the Stein analogue of the O-S-T

The referee has suggested

If c:,j=l

4 becomes

3 and 5 become

4

corollary

Since the matrix D whose elements

eigenvalue

and all other eigenvalues

5 and Carlson’s

zero,

Theorem.

ACKNOWLEDGMENT The results of this paper form a major part of the author’s

Ph. D. Thesis

[3]

written at Oregon State University under Professor David Carlson. The author wishes to express his gratitude to Professor Carlson for the help and encouragement which he has given throughout Linear

the development Algebra

of these results.

and Its Applications

2(1969),

131-142

142

I<. D. HILL

REFERENCES 1 M. P. Drazin, Some generalizations

of matrix commutativity,

Proc. London

Math.

Sec. 1(1951), 222-231. 2 M. P. Drazin, J. W. Dungey, and K. W. Gruenber,,= Some theorems on commutatix matrices,

J. London Math. Sot. 26(1951),

3 R. D. Hill, Generalized Inerlia Theovy State University,

221-228.

for Complex

Ph. I>. Thesis, Oregon

Matvices,

1968.

4 M. Marcus and H. Mint, A Survey

01 Matrix

Theory

and

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