STABLE MATRICES, THE CAYLEY TRANSFORM, AND CONVERGENT MATRICES TYLER

VOL. 14 NO. (1991) 77-82

STABLE MATRICES, THE CAYLEY TRANSFORM, AND CONVERGENT MATRICES TYLER

HAYNES

Mathematics

Department

Saginaw

Valley State University

University Center, Michigan48710

(Received January 17, 1990)

ABSTRACT.

The main resultis thata

square

^matrix

D

isconvergent

(lim D" 0)

^ifand

onlyifit is the Cayley transform

C^ (I-A)" (I+A)

of a stablematrixA,where astable matrix is onewhose characteristic values all havenegative real parts.

In

passing, theconceptof Cayleytransformis generalized, and thegeneralizedversionis showncloselyrelated to the equation AG

+

^GB

^D.

^Thisgives riseto acharacterizationof the non-singularityofthe mapping

X AX + XB.

As

consequences

arederivedseveralcharacterizations ofstability (closelyrelated to

Lyapunov’s result)

which involveCayleytransforms.

KEY

WORDS

AND PHRASES.

Stable matrix,Cayley transform, convergent ^matrix.

1980AMS SUBJECT

CLASSIFICATION CODES.

15A04, 15A24

BothTausskyand Stein

[Stein, 1965]

^{havewrittenonthe} connection between stable matrices andconvergent matrices. The linkjoining thetwo is theCayley^{transform: a matrixis} convergent it is the

Cayley

transformofastable matrix

(theorem 8).

Cayley

transforms are introducedbyconsideringthe matrixequation

AX+XB

C. But firstalemma:

Lemma 1" Over field

F

let matrix

A

be nxnand letxbeeither indeterminate overF orin

F

but not a characteristicvalueof

A.

Then

(xl-A)-’(xl + A) (xl +A)(xI-A)k (I)

If either expression in

(1)

^isdenoted by

CAx

^then

^C A,=C,.

^Ifx

^,,

^{0, then}

A x(CAcI)(CA,+ ^I)

^1.

⁽²⁾

Proof: Sincexis not acharacteristicvalue ofA,

(xI-A)

^exists.

(1)

follows from

(xl +A)(xI-A) (xI-A)(xl +A). (3)

Before

(2)

canbederived, the non-singularityof

CA, ⁺

^{mustbe proven. This}

equationholds:

CA, ⁺ (xI+A)(xI-A)" + (xl-A)(xI-A)"

2x(xl-A) -.

Therefore,

CA ^{+ I[} 2xlxI-Al" ,

0since x

,

0and

Ix ^{I-A[ ,}

⁰

^{(for xI-A}

^is non-singular);

hence,

CA, ⁺

^is non-singular.

(2)

thenfollows directly. ^QED

CA.

^of

⁽¹⁾

^{is the}generalized

Cay!ey

transformof

A.

Ifx is not acharacteristic value of

A,

then

CA,

^isthe

^Cayley.

^{transform of}

^A;

^{itwillbedenoted}

C^.

^{Note that the}

mapping

A-*C^

^is bijective from thesetof matriceshavingno characteristic value onto those havingnocharacteristic value -1, the inverse transformationbeingdetermined by

(2).

Theorem

2":

Letmatrix

A

be mxm,Gand

D

be mxn, and

B

be nxn, allwith entries in field F.

AG+GB

D G-CA’XGCB,

x

-2x(xI.-A)lD(xI.-B) 1, (4)

wherexis either indeterminate overForin

F

but

,.,

0and acharacteristic value ofneither

A

norB.

Proof: xsatisfies the requirementsfor

CA,

^and

Ca"

x toexist, according to the lemma, and the dimensions of

CA, Ca,x, (xtm-A)",

^and

(xI,-B)

^{a are}such that theexpressionon the rightof

(4)

^iswell-defined.

AG+GB

D

(xO-AO)(xl.-B) (xG +AG)(xI.+B)

^-2xD

(xl.-A)G(xI.-B) (xI. +A)O(xI. +B)

^-2xD

G-(xI.-A)"(xI(R) +A)G(xI. + B)(xI.-B)

^*

-2x(xI.-A)ID(xI.-B)

G-CA, xocB; -2x(xl.-A)"D(xI.-B )"

^QED One consequence^ofthe

preceding

theorem is the celebrated result that every properly

orthogonal’"

^matrixPcan beexpressed^as

P (I+K)(I-K),

^where

^K

^{is a real skew matrix. To}

derive it, inthe theorem let

F

realnumberfield, G

I, D

O,x -1, and B A’.

Then itfollows that

A+A’

0

PP’ I,

where

P (-I-A)(-I+A) (I+A)1(I-A),

the relationshipbetween

P

and

A

beingdeterminedby

(1)

and

(2)

of the lemma

(cf.

^{the remark}

on thebijectivecharacter of

A-,C^).

Likewisethe

Cayley parametrization

ofunitarymatrices follows

[Gantmacher,

Vol.

I;

p. 279

(95)].

Overa field

F

let

A

be anmxm matrix,

X

an mxn matrix and

B

an nxn matrix. Let

ZA,

B

AX +

XB. Clearly themapping

ZA, B: X-AX + XB

^{is a linear}transformation on the

"This theoremgeneralizesalemma of

Weyl’s [Weyh p. 57,

lemma

(2.10.A)].

"’An

orthogonai matrixisproper noneofits characteristicvalues -1.

linear spaceofmxn matrices. Denote

.tA,

A.by

.tA: .tA(X) AX +

XA*, whereall matrices arc of thesame dimension.

Corolla

^{3: Let}

^{A, B, G,}

^{x, and}

^F

^{be as}in theorem 2. Then themapping

G-G-C^.xGCaa

^{is linear from theset}of all mxn matrices intoitself. This mappingis non- singular

,t:A,

^is non-singular.

Proof: Thelinearityofthe mappingis obvious.

,tA,

a^isnon-singular forcvery

D

thereexists asolution of

AX+XB D

^=,for

every E

there exists a solution of

X-CA.xXCa

E (thcorem

^{2 and the}non-singularityof

xI,-A

and

xI-B)

^themapping

G-,G-C^.xGCt.

^is

non-singular. QED

In

therestof this article, let

F

be the field ofcomplexnumbers and let all matrices be square.

The inertia of an nxn matrix

X

is the orderedtripleofintegers

0r(X), ,(X) 6(X)) In(X),

where

n(X)

isthe numberof characteristic values of

X

whose realpartsare positive, ,(X), the numberwhose real parts arenegative, and

a(X)

the number whose real parts are 0.

Corollary^4: If

A

hasnocharacteristic value =1, then

In(I-CACA*) In(-(A+A*)).

Proof:

CA. CA*

^byaslight modification of lemma 1.

In

theorem2, let B A*, G

I,

and x 1; thcn

D

A+A*. Therefore,

I-CACA* I-CAIC

A.

-2(I-A)1(A+A*)(I-A*)

¹

(I-A)"[-2(A+A*)][(I-A)1]

^{*. Sincethe lastexpressionis}congruent to

-2(A+A*),

^their

inertias arethe same, and

In(-2(A+A*)) In(-(A+A*)).

^QED

A

squarematrix isstable all its characteristic values havenegativereal parts. S denotesthe set ofallstable nxn matrices,IIdenotes thesetofallpositive-definite^hermitian matrices and h/denotcs thesetofallnegative-definitehermitian matrices.

Theorem 5:

A

S for

any G,II

there existsG II

G-CAGCA* G

thereexists

G1II G-CAGCA* G1

^{forsomeG rl.}

Proof:

In

theorcm 2, let

B

A*,x

(for

^{is not}characteristicofastable matrix and

CA

presupposes thatx

,, _1),

and

D -Y2(I-A)G(I-A*).

Then the last termof

(4)

^is

G,

and

(4)

becomes

AG+GA*

D G-CAGCA*

D is hermitelycongruent to-V2G, andso

In(D) In(-V2G 0.

^Therefore,

GII

^D

Firstequivalence: Assume

A

S. For any

G,II, D

tO. Therefore, ^"qG rI:AG+GA*

D

[Taussky],^so

G-CaGCA* G.

Conversely,if for any

GII

^{thereexistsG rI:}

G-C,xGCA* G,,

thcn AG+GA*

D;

since

Gt

^is

^arbitrary,

^sois

^D,

^for

^I-A

^and

^I-A*

^{are non-}

singular,otherwise

C^

^and

Ca* CA.

^{would not}be defined. Since

D tO, A

S[Taussky].

Second equivalence: Assume

A

S. Then 3G elI:AG+GA* D forsome D andso

G-C^GC^* ^GI; Gleli

^{as above.}

Conversely,

if, for someGtelI,

G-C^GC^* G1

^for

someG eli, thenAG+GA* DandD eh/.

Hence, A

eS. QED

Corollary6:

A

S -qG eli: I-diag(g

g,)

eli,where

{g}

are theroots of

xG-C^GC^*

^0;furthermore,

&

^{is real}

(i=l n).

Proof: Assume

A

S.

By

the firstequivalenceof the precedingtheorem 3G eli:

G-C^GC^* ^I.

^{Since bothGand}

C^GC^*

^arehermitian andG eli,3R: Ris non-singular and R’GR

I, R’(C^GC^*)R

^{diag(g g,,)where}

{g}

^arethe rootsof

xG-C,,GC^*I

^0.

Then

R’R R’IR R’ (G-C^GC^*)R R’GR-R’(C^GC^*)R=

I-diag(g g.). R’

R

eli

becauseR’GR

R"IR "

^{G eli}

^RR’

^li

^R’R

^eli. Therefore, I-diag(g, g.) II.

Since Gand

C^GC^*

are hermitian and G eli,3R:

R

is non-singularand R’GR I,

R’(C^GC^*)R

diag(gt

g,,)

where

{g}

are the

(real)

^{roots of}

xG-C^GC^*I

^{0. Then}

R "t[I-diag(g

,g,)]R^q

R "RI-R "ldiag(g g,)R

¹

G-C^GC^*

^eli.

^By

^{the secondequiva-}

lenceofthepreceding theorem,

A

S.

g, isreal (i=

n) [Gantmacher,

Vol.

I;

p. 338, thm.

22].

^QED

Corollarvfl7: ^A

^{eS "qG eli: g <}

(i=l ,n)

where

{g}

arethe characteristic valuesof

G’C^GC^

^*.

Proof:

In

the preceding corollary, Gis non-singularsinceG ell. Hence,

{g,},

the roots of

xG-C^GC^*l

^0,a the characteristicvalues of

GtC^GC^

^{*, for}

^{I,G-C^GC^}

GI. x-G-C^GC^*l

^{0. I-diag(gt g,,) eliisequivalent to i-g, > 0(i-1}

^n).

^QED

Thealgebraic propertiesoftheCayleytransformpreviously developedwillbe applied to

prove

theorems about convergentmatrices.

The nxn matrix

A

isconvergen.t lim

A"

0.

m-,

Theorem8:

D

is convergent^,=,3

A

S

D Ca.

Proof:

D

is convergent D* is convergent.

Assume

that

D

isconvergent. ThenD*is convergent.

By

Stein’stheorem [Stein, 1952; p. 82, thm.

1] (-G eli)(-GelI)

G-DGD*

Gt.

^Define

^A

by

A (D-I)(D+I);

^then

D Ca. ^By

theorem 2, AG+GA*

-1/2(I-A)G(I-A*).

^Since

-1/2(I-A)G(I-A*)

^is hermitely congruentto

-G,

AG+GA* h/andby

[Taussky] A

Assume

that

A

S. Then

by

theorem

5, (qG elI)(qGII): G-C^GC^* G. ^By

Stcin’stheorem,

Ca*

^is convergent, ^andso

C^

^is convergent. ^QED Corollary^9: D isconvergent

(vGtII)(3GalI):

G-DGD*

G,

(3Gt,II)(IG,II):

^G-DGD*

Gt.

Proof:

By

the preceding theorem,

D

isconvergent

D

C^, where

A

S. Thetwo

equivalences follow from this fact and theorem 5. QED

Thepreceding

corollary

is a theoremofTaussky’s [Taussky; p. 7, ^thm.

5],

which isitself astrengthening of Stein’s theorem.

REFERENCES

Gantmacher, F. R. The

Theo_ of

^Matrices. 2 vols. Translatedby K. A. Hirsch. New York: Chelsea Publishing

Company,

1960.

Stein, P. "SomeGeneral Theoremson Iterants." Journal

of

^Research

of

^theNational

Bureau

of

^{Standards,Vol. 48, No.}

⁽¹⁹⁵²

^{January), 82-83.}

Stein, P. "On the

Ranges

ofTwoFunctions ofPositive Definite Matrices." Journal

of

Algebra,Vol. 2, No.3

(1965

September)350-353.

Taussky, Olga. "MatricesCwith C"-,0." Journal

of

^{Algebra,Vol. 1, No.}

⁽¹⁹⁶⁴

^April),

5-10.

Weyl, Hermann. TheClassical

Groups:

Theirlnvariantsand Representations. 2nd ed.

Princeton, N.J.: Princeton UniversityPress, 1946.