SEMI STABILITY FACTORS AND SEMI FACTORS. Daniel Hershkowitz and Hans Schneider

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1 Cntemprary Mathematics Vlume 47, 1985 SEMI STABILITY FACTORS AND SEMI FACTORS * Daniel Hershkwitz and Hans Schneider ABSTRACT. A (semistability) factr [semifactr] f a matrix AE r: nn is a psitive definite [psitive semidefinite] matrix H such that AH + HA* is psitive semidefinite. We give three prfs t shw that if A has a semistability factr then it cannt be unique. We give necessary and sufficient cnditins fr a matrix H t be a (semi)factr f a given matrix. We als determine the dimensin f the cne f semistability factrs. 1. INTRODUCTION. A (square) cmplex matrix A is said t be (psitive) stable [semistable] if all its eigenvalues lie in the pen [clsed] right halfplane. The matrix A is said t be near stable if A is semis table but nt stable. A very well knwn characterizatin f stable matrices is essentially due t Lyapunv [8], see als [5, vl. II, p. 189], [9], [10], [4] and the references there. THEOREM 1.1 (Lyapunv-Gantmacher): A cmplex matrix A is stable if and nly if there exists a psitive definite (hermitian) matrix H such that ( 1.2) AH + HA* > O. We have here written H > 0 t dente a psitive definite (hermitian) matrix. We als write H > 0 t dente a psitive semidefinite matrix. Fr near stable matrices the situatin is mre cmplicated, as shwn by the fllwing result [4, Crllary III.1]. THEOREM 1.3. Let A be a cmplex matrix. Then there exists a psitive definite matrix H such that ( 1.4) AH + HA* > O. if and nly if A is semis table and ( 1.5) the elementary divisrs f all pure imaginary eigenvalues f A are linear. * Research f this authr was supprted in part by NSF grant DMS American Mathematical SOciety /85 $ $.25 per page

2 204 HERSHKOWITZ & SCHNEIDER We shall call the matrix H in (1.2) r (1.4) a semistability factr f the matrix A. An interesting special case ccurs when a semistability factr may be chsen t be diagnal. In this case we call the matrix Lyapunv diagnally stable [semistable] if there exists a psitive definite diagnal H satisfying (1.2) [(1.4)], and the crrespnding diagnal H is called a Lyapunv scaling factr f A, e.g. [6] and [7]. At the Bwdin meeting n "Linear algebra and its rle in systems thery" (July 1984) ne f us discussed necessary cnditins and sufficient cnditins fr the uniqueness (up t psitive scalar multiplicatin) f a Lyapunv scaling factr f a given Lyapunv diagnally semistable matrix (see [6]). At the end f the talk Dale Olesky raised the questin whether there are cases where a semistability factr f a given semis table matrix f rder greater than is unique. In this paper we shw that the answer t that questin is negative and we give three different prfs. Our third prf is a cnsequence f results that g cnsiderably beynd the questin f uniqueness f stability factrs. If A r: nn we call a matrix H a (semistability) semi factr f A if H > 0 and (1.4) hlds. In Sectin 4 we determine necessary and sufficient cnditins fr H t be a semifactr. We give ur principal results in tw frms. The first (Therem 4.7) is fr a matrix A in blck diagnal frm. The secnd (Therem 4.12) is an invariant frm which hlds fr all A r; nn. We shw that ur therem generalizes a part f Therem III f [4] which characterizes the ranks f semifactrs f A. In Sectin 5 we define the (semistability) factr cne S(A) and the semi factr cne Se(A) f A E r;nn. We shw that SO(A) is the clsure f the set f semifactrs f A f maximal rank (Lemma 5.2) and we deduce a necessary and sufficient cnditin n A fr SO (A) t be the clsure f S (A) (Crllary 5.9). We als give a frmula fr the dimensin f S O(A) (Therem 5.10) and as a crllary (Crllary 5.11) we btain ur third prf f the nnuniqueness f semistability factrs. Our paper is related in spirit t the papers by Carlsn [3] and Avraham-Lewy [1], thugh there is little verlap in results. Thse papers discuss the general prblem f the relatin f ranks and inertias f A C nn and the hermitian matrices Hand K = AH + HA*, in [3] under the assumptin that K > 0 and in [1] under the assumptin that H > O. 2. STATEMENT OF THE THEOREM AND PROOF NO.1. THEOREM 2.1. Let A be a cmplex matrix f rder greater than 1. Then either A has n semistability factr r A has at least tw linearly inde-

3 SEMI STABILITY FACTORS AND SEMIFACTORS 205 pendent semistability factrs. PROOF. Suppse that A has at least ne semistability factr. Then, by Therem 1.3, A is semistable. Suppse that A is stable. By Therem 1.1, A has a semistability factr H such that (1.2) holds. Let K be any psitive definite matrix such that Hand K are linearly independent. Since the psitive definite matrices frm an pen set in the tplgical space f hermitian matrices, it fllws that fr sufficiently small psitive E, Hand H + EK are linearly independent semistability factrs f A. Nw suppse that A is near stable. Then A has a pure imaginary eigenvalue A = ia. Let x be an eigenvectr f A assciated with the eigenvalue A. S, (2.2) Ax = iax and by multiplying bth sides f (2.2) by x * we btain (2.3) Axx * iaxx*, which is equivalent t (2.4) -iaxx*. It fllws frm (2.3) and (2.4) that (2.5) D. Let H be a psitive definite matrix such that (1.4) hlds. Observe that H' = H + xx* is als a psitive definite matrix and, further, by (2.5) and (1.4), H' is a semistability factr f A. Since H is nnsingular and since xx * is a matrix f rank it fllws that xx * is nt a scalar multiple f H, and therefre H' is nt a scalar multiple f H. Hence, H and H' are tw linearly independent semistability factrs f A. [] 3 PROOF NO.2. We begin with a simple lemma. LEMMA 3.1. Let B,H I! n,n such that H is a psitive definite matrix. If BHB* = D then B = D. PROOF. Given that BHB* D, it fllws that (3.2) (B*X)*H(B*X) D, 'iixej[;n.

4 206 HERSHKOWITZ & SCHNEIDER Since H is psitive definite it follws frm (3.2) that B*x = 0, VXElC n, which implies B* and B = O. LEMMA 3.3. Let B be a semis table matrix, let H > 0 and let a be any real number. Then H is a semistability factr f B if and nly if H is a semistability factr f B + iai. ~. Observe that BH + HB* = (B + iai)h + H(B + iai)*. Prpsitin 3.4. Let B be a singular semis table matrix. Then B des nt have a unique semistability factr. ~. If B then the claim is clear. S we may assume that (3.5) B '* O. Assume that B has a unique semistability factr H. Thus (3.6) C BH + HB* > O. Clearly, (3.7) BCB* > O. Since H is psitive definite, s.is H" = H + BHB*, and by adding (3.6) and (3.7) we btain that H" is a semistabili ty factr f B. Our uniqueness assumptin yields that H" is a scalar multiple f H, and hence BHB* is a scalar multiple f H. Since H is nnsingular and B is singular it fllws that BHB* = O. By Lemma 3.1, B = 0, which is a cntradictin t (3.5). Hence, the uniqueness assumptin is false. PROOF NO.2. As in the first prf we may assume that A is a near stable matrix. As such, A has a pure imaginary eigenvalue. S, A + iai is singular fr sme real number a. Our therem nw fllws frm Prpsitin 3.4 and Lemma SEMIFACTORS. LEMMA 4.1. Let A E IC nn be such that the spectrum f A cnsists f a single pure imaginary eigenvalue A, and let H > O. Then H is a semista-

5 SEMI STABILITY FACTORS AND SEMIFACTORS 207 bility factr f A if and nly if (4.2) (A - U)H = O. PROOF. If (4.2) hlds then AH = AH and (1.4) fllws. Cnversely, suppse that (1.4) hlds. Let U be a unitary matrix such that A' = U*AU is upper triangular. Let H' be the psitive definite matrix U*HU. Dente by B the matrix A' - AI. Observe that BH' + H'B* > O. Clearly, the last rw f BH' is zer. Assume that BH' * 0 and let K be the index f the last nnzer rw f BH'. Partitin B by B where B11 is k x k, and partitin H' and BH' cnfrmably. Since [BH')22 = 0 and [BH')21 that [BH')12 = O. Hence it fllws frm the fact that BH' + H'B* > 0 (4.3) O. Since (4.4) it fllws frm (4.3) that (4.5) It is a well knwn prperty f psitive semidefinite matrices, e.g. [7), that (4.5) implies (4.6) O. Therefre, by (4.4) and (4.6), we have ROW k [BH')'1 = (ROW k B 11 )H;1 + (ROW k B 12 )H;1 = O. Since [BH')12 = 0 it fllws that Rwk(BH') = 0, which is a cntradictin. Hence, BH' = 0 and s (A - U)H O.

6 208 HERSHKOWITZ & SCHNEIDER THEOREM 4.7. Let where A q+2 q. A :c nn.;::j i i=1 0, i 1,,q and ~ i, j ~ q. where Aq+1 and -Aq+2 are stable. Let H > 0 be partitined cnfrmably. Then H is a semifactr f A if and nly if (4.8) where (4.9) H q+2 B Hii i=1 0, i 1,,q, (4.10) and (4.11) * A H + H A > 0, q+1 q+1,q+1 q+1,q+1 q+1 - Hq +2,q+2 =. PROOF. If H satisfies (4.8) - (4.11) then clearly AH + HA 2- O. Cnversely, let AR + HA* 2-0 fr sme H 2-. By Lemma 4.1, cnditin (4.9) hlds. Cnditin (4.10) is bvius and by Lemma 1 f [2], we btain (4.11). T cmplete the prf we must shw that Hij = 0, i '* j, i,j = 1,,q + 2. If i = q + 2 then Hij = 0 and Hji j = 1,,q + 1, since Hii = 0 and H 2- O. Let i '* j, Withut lss f generality we may assume that 1 < i ~ q. By (4.9), 0, 1 < i, j~q + 1 Since AH + HA* > 0 it fllws that * (AR + HA ).. 1).

7 SEMI STABILITY FACTORS AND SEMIFACTORS 209 Hence, by [5, Vl. I, p. 220], 11 spec(aj). Hij since Ai + ~ * 0, whenever Let A r: nn and let a be a cmplex number. We define the eig:ensl2ace Ea(A) as the nullspace f (A - ai) and we define the 2eneralized eig:ensl2ace Ga(A) as the nullspace f (A - ai)n. Observe that Ea(Al {O} and * Ga(A) {O} if and nly if a is an eigenvalue f A. We dente by G+(A) * the vectr space sum ~ a spec(a) Re a>o G (A). a The fllwing therem is an invariant frmulatin f the results f Therem 4.7. THEOREM Let A r: nn and let H > O. Then H is a semi factr f A if and nly if (4.13) H L Ha + H+, a spec(a) Re a>o where (4.14) (4.15) and (4.16) PROOF. First suppse tha t A satisfies the hyptheses f Therem 4.7. Suppse that AH + HA* > O. Then, by Therem 4.7, H satisfies (4.8) - (4.11). Let q+2 (4.17) HA e ijhjj i i=l i 1, q, where ij is the Krnecker delta, and let q+2 (4.18) H e 0 q+1, jhjj + j=l

8 210 HERSHKOWITZ & SCHNEIDER Then H has the frm (4.13). Observe that (4.14) fllws frm (4.9) and (4.17), and that (4.15) and (4.16) fllw frm (4.10) and (4.18) since G+(A) is spanned by the clumns f q+2 e j=1 I q+1,j Cnversely, assume that (4.13) - (4.16) hld. In view f the frm f A it fllws that there exist Hii, i = 1, q + 1 satisfying (4.8) - (4.11). Hence, by Therem 4.7, AH + HA* ~ O. Nw suppse that A is a general matrix in 1: nne Then there exists a nnsingular matrix T such that A' = TAT-1 satisfies the hyptheses f Therem 4.7. Let H' THT*. Then clearly AH + HA* > 0 if and nly if (4.19) A'H' + H' (A')* ~ O. By the discussin abve, (4.19) holds if and nly if H' satisfies where H' = I exe special Re ex=o H' + H' ex + (4.21 ) (4.22) and Since fr any matrix B and since (4.20) - (4.23) are equivalent respectively t (4.13) - (4.16) whenever and

9 SEMI STABILITY FACTORS AND SEMIFACTORS 211 Remark. Observe that (4.14) is equivalent t AHa = aha. COROLLARY Let A E lc nn and let H ~ O. If AH + HA* 2. 0 then PROOF. By Therem 4.12, H satisfies (4.13) - (4.16). Hence 0, a E spec(a), Re a 0, and it fllws that (4.25) AH + HA* Since it fllws frm (4.15) and (4.25) that The fllwing crllaries are derived either frm Therem 4.7 r Therem COROLLARY Let AE~,n. Then -A is semis table if and nly if H > 0 and AH + HA* > 0 imply that AH + HA* = O. PROOF. If -A is semistable then G+(A) = {O} and ur claim fllws frm Crllary Cnversely, if -A is nt semis table then let a E spec(a) such that Re a > O. Let x be an eigenvectr f A assciated with a. Observe that the psitive semidefinite matrix H = xx* satisfies AH + HA* = 2 Re(a)xx* which is a nnzer psitive semidefinite matrix. COROLLARY Let B E r: nne Then B has n pure imaginary eigenvalue if and nly if K > 0 and BK + KB* imply that K = O. PROOF. There exists a nnsingular T such that A = TBT- 1 satisfies the hyptheses f Therem 4.7. It is enugh t prve that A has n pure imaginary eigenvalue if and nly if H > 0 and AH + HA* = 0 imply that

10 212 HERSHKOWITZ & SCHNEIDER H =. If A has an imaginary eigenva1ue and x is a crrespnding eigenvectr, then xx* '" 0 and A(xx*) + (xx*)a * =. Cnverse1y, suppse that has n pure imaginary eigenva1ue and that H > 0 satisfies AH + HA* =. Then A = 1'.1 e 1'.2' where the spectrum f 1'.1 and f -1'.2 are in the pen right ha1fp1ane. By Therem 4.7, H Hl1 0 H22, where H22 = 0 and A1Hll + Hll 1'.1 * =. But, since a + F '" 0 fr every a,fl spec(al), it f11ws frm [5, V1. I, p. 220] that Hll =. Hence H =. 0 A As a direct cnsequence f (4.13) - Therem III f [4]. (4.15) we btain the first part f COROLLARY Let A :c nne If H > 0 and AH + HA* > 0 then (4.29) rank H < peal + 1f(A), where peal is the number f e1ementary divisrs f the pure imaginary rts f A and 1f(A) is the number f eigenvalues f A (cunting mu1tiplicities) which lie in the pen right halfplane. The fllwing immediate cr1lary strengthens Cr11ary f [4], (ur Therem 1.~ by characterizing semistability factrs. COROLLARY Let A!C nn and 1et H > O. Then H is a semistabi- 1ity factr f A if and nly if A is semis table, (1.5) holds and H satisfies (4.13), (4.16) and Range Ha Range H+ 5. FACTOR CONES AND SEMI FACTOR CONES. Let H be the vectr space ver the rea1 fie1d cnsisting f cmplex n x n hermitian matrices. As in [2, p. 2] a cne C in H is a nnempty subset H f which is clsed under additin as we11 as nnnegative sca1ar mu1tiplicatin. By span C we dente the subspace spanned by C and we wri te dim C fr the dimensin f span C The (semistability) ~ ~ SeAl is defined by seal = {O} U {H > 0 : AH + HA* ~ O} and the (semistability) semifactr cne S (A) is given by SO (A) = {H ~ 0 AH + HA * > O}.

11 SEMI STABILITY FACTORS AND SEMIFACTORS 213 We bserve that the clsure f SIAl (in the usual Euclidean tplgy), dented by cl SIAl, satisfies (5.1) cl SIAl I: SOIA). The reverse inclusin des nt necessarily hld as demnstrated by By Therem 1.3 we have SIAl {O}, while the psitive semidefinite matrix, satisfies AH + HA* > O. Hwever, let rank H k}, and let Observe that by Therem III f [4] the set ~ (A) if 0 ~ k~ rae Then we have: is nnempty if and nly LEMMA 5.2. Let k be a nnnegative integer. Then -%(A) = cl f"(a) if and nly if k = rae PROOF. Clearly S (A) is a clsed set. Hence, fr all k, cl T~A) ~ SO(A). If k < ria) then, since the limit f rank k matrices has rank less than r equal t k, and since SOIA) cntains a matrix f rank ra, it fllws that SOIA) * cl ~(A).

12 214 HERSHKOWITZ & SCHNEIDER If k = r(a) then let H SO(A) and chse K ~ (A). Then K has a psitive definite principal submatrix f rder ra' It fllws that the crrespnding submatrix f H + sk is nnsingular fr all s > O. Hence ( 5.3 ) rank(h + sk) ~ ra' By the definitin f RA' equality holds in (5.3). Thus H cl ~ (A). 0 LEMMA 5.4. Let k be a nnnegative integer. Then (5.5) SO(A) = cl( {O} U T~(A» if and nly if either S O(A) = {O} r k = ra' PROOF. TttA) "*, If T~ (A) = ~ then (5.5) hld if and nly if S (A) = {O}. If then cl({o} UT~(A» = cl ~ (A) and by Lemma 5.2, (5.5) hlds if and nly if k = ra' PROPOSITION 5.6. Let A be a cmplex n x n matrix. Then (5.7) S O(A) cl S (A) if and nly if either -A is stable r A is semistable and the elementary divisrs f all pure imaginary eigenvalues f A are linear. PROOF. By Therem III f [4], (5.8) ra = n(a) + p(a), (where n(a) and p(a) are defined in Crllary 4.28). Since S(A) = {O} U T~ (A) it fllws frm Lemma 5.4 that (5.7) hlds if and nly if either SO(A) = {O} r ra = n. By (5.8), SO(A) = {O} if and nly if -A is stable, and, by Therem 1.3, ra = n if and nly if A is semis table and (1.5) holds. COROLLARY 5.9. Let A be a cmplex matrix. If S(A) "* {O} then cl S(A) = SO(A). PROOF. If S(A) "* {O} then ra = n and since S(A) claim fllws frm Lemma 5.4. THEOREM Let A r: nn. Then dim SO(A) = 21 [n(a)(n(a) + 1) + L " ae spec(a) Re a=o

13 SEMISTABILITY FACTORS AND SEMIFACTORS 215 where ea(a) is the dimensin f the eigenspace Ea(A). ~. Withut lss f generality we may assume that A is in its Jrdan cannical frm and satisfies the hyptheses f Therem 4.7. By Therem 4.7, SOIA) cnsists f all psitive semidefinite matrices H fr which (4.8) - (4.11) hld. Let 1 ~ i ~ q. By (4.9), Hii is any psitive semidefinite matrix all f whse entries are 0, except pssibly thse which lie in the intersectins f first rws and first clumns f Jrdan blcks which crrespnd t the eigenvalue Ai' The number f such rws (clumns) is eai(a). Hence the number f such linear independent Hii is (eai(a)(eai(a) + 1) /2. We must still shw that the number f linearly independent psitive semidefinite matrices H q +l,q+l which satisfy (4.10) is n(a)(n(a) + 1)/2. By Therem 1.1 there exists a psitive definite matrix H q +l,q+l such that A H + H A * > O. q+l q+l,q+l q+l,q+l q+l Hence by cntinuity arguments, there exists an pen set f such matrices which satisfy (4.10). As is well knwn, the span f these matrices (ver the real field) is the whle set f hermitian matrices f rder n(a). fllws. Our therem nw COROLLARY (and prf N.3). Let A e C nn, n > 1, be such that SIAl * {Ole Then dim SIAl = dim SOIA) > 1. ~. Since SIAl * {O} it follws that ra = n and als, by Crllary 5.9, cl SIAl = SOIA). Hence, as is well knwn, dim SIAl = dim SOIA). hlds: By (5.8) ne f the fllwing three pssibilities (i) n(a) ~ 2. ( ii) n(a) and pia) ~ 1. (iii) n(a) and pia) ~ 2. In each case dim SOIA) > by Therem We remark that using essentially the same arguments we can shw that there exist at least tw linearly independent semifactrs f A prvided that ra > 1. f rank ra

14 216 HERSHKOWITZ & SCHNEIDER BIBLIOGRAPHY 1. AH + HA* S. Avraham and R. Lewy, On the inertia f the Lyapunv transfrm fr H > 0, Lin. Mult. Alg. 6(1978), A. Berman and R. J. Plemmns, "Nnnegative matrices in the mathematical sciences", Academic Press, N.Y D. Carlsn, Rank and inertia bunds fr matrices under R(AH) ~ 0, J. Math. Anal. Appl. 10(1965), D. Carlsn and H. Schneider, Inertia therems fr matrices: the semidefinite case, J. Math. Anal. Appl. 6(1963), F. R. Gantmacher, "The Thery f Matrices", Vls. I,ll. Chelsea, New Yrk, D. Hershkwitz and H. Schneider, Scalings f vectr spaces and the uniqueness f Lyapunv scaling factrs, Lin. Mult. Alg. (t appear). 7. D. Hershkwitz and H. Schneider, Lyapunv diagnal semistability f real H-matrices, Lin. Alg. Appl. (t appear). 8. A. Lyapunv, Prbleme General de la stabilite du muvement, Cmmun. Sc. Math. Kharkv (1892, 1893); Annals f Mathematical Studies, Vl. 17, Princetn Univ. Press, Princetn, N.J., A. Ostrwski and H. Schneider, Sme therems n the inertia f general matrices, J. Math. Anal. Appl. 4(1962), O. Taussky, A generalizatin f a therem by Lyapunv, J. Sc. Ind. Appl. Math. 9(1961), DEPARTMENT OF MATHEMATICS UNIVERSITY OF WISCONSIN MADISON, WI 53706