Stability and D-stability

Stability and PStabllity* Biswa Nath Datta lnstituto de Materndtica, Estutistica e Ci&ncia o!a ComputaqZ Universidah Estuduul de Campinas Caixa Postal 1170 13.100 Campinus, S.P., Brad

Submittedby D. Carlson

ABSTRACT In this paper, we prove a new sufficient condition for D-stability of a matrix. We then use it to show that (i) a nonderogatory stable matrix A has a Schwarz canonical form which is D-stable, and (ii) a nonderogatory stable matrix A has a Routh-canonical form which is totally stable.

1.

INTRODUCTION

A real matrix A is called a stable matrix if all the eigenvalues of A have negative real parts. A is D-stable if DA is stable for every choice of a positive diagonal matrix D. The concept of D-stability is an important concept in mathematical economics [l, 8, 121. No necessary and sufficient condition for D-stability is yet known, though effective characterizations of H-stability in case A is a complex matrix and of S-stability in case A is a real matrix have been obtained by Carlson [3], by Carlson and Schneider [4] and by Ostrowski and Schneider [ll]. The strongest sufficient condition for D-stability known so far is the following [l] : Given A, if there exists a positive diagonal matrix H such that HA+ATH=

W,

where W is negative definite, then A is D-stable. ‘Presented in “287 Redo to 14 July 1976, Bra&a.

Anualde SociedadeBrasileirapara o progress0da C&n&“, 7

LINEAR ALGEBRA AND IIS APPLICAZYZONS 21, KG-141 (1978) 0 Elsevier North-Holland,Inc., 1978

135

0024-3795/78/0021-0135$01.25

BISWA NATH DATI’A

136

Clearly, every D-stable matrix is stable, but the converse is not ture. A natural question therefore arises: when does stability imply D-stability? This question was considered by the economists Arrow and McManus [l] and Enthoven and Arrow [B], and by the mathematician Taussky [14, 151. It is known that (i) a stable matrix with non-negative off-diagonal elements and negative diagonal elements is D-stable, and (ii) the negative of a dissipative matrix is D-stable (a matrix A is called dissipative if it is the negative of a stable matrix and is such that A + A r is positive definite [ 151). Taussky [15] showed that the matrix S - kl, where S is a real skew symmetric matrix and k is a real positive scalar, is D-stable. Recent work of Johnson [9, lo] has identified several other classes of D-stable matrices. In this note, we add a few more classes to this list. We prove that (a) a nonderogatory stable matrix A in its Schwarz canonical form is D-stable, and (b) a nonderogatory stable matrix in its Routh canonical form is totally stable (a matrix A is called totally stable if every leading principal minor of A is D-stable [2, p. 1161. We note that the sufficient condition for D-stability given by Arrow and McManus and cited above can be improved by using a recent inertia theorem due to Chen [5] and Wimmer [16]. We use this improved sufficient condition to prove (a) and (b). Finally, given a stable Schwarz matrix S, we construct a matrix S’, similar to S, such that S’ is totally stable.

2.

AN INERTIA

THEOREM

The inertia of a matrix A is defined to be a triplet (n(A),p(A),G(A)), where n(A), V(A) and 6 (A) are respectively the numbers of eigenvalues of A with positive, negative and zero real parts. It is denoted by In(A). Let the nor? matrix (N,AN,A’N,...,A”-‘N) be denoted by [A,N]. Then the following theorem has been recently proved by Chen [5] and independently by Wimmer [16]. THEOREM 1. Let A be an n X n matrix, and let there exist a symmetric matrix X such that the matrix N given by AX+XAT=N, where AT is the transpose of A, is positive semi&finite rank[A, N] = n. Then 6 (A) = 0 and In(A) =In(X).

0) and such that

137

STABILITY AND D-STABILITY 3.

A SUFFICIENT CONDITION

FOR D-STABILITY

THEOREM 2. Let A be an n X n real upper [lower] Hessenberg matrix with non-zero subdiagonal [superdiagonal], and let there exist a symmetric matrix H such that

AH+HAT=

- W,

(2)

where W is a positive semidefinite matrix having a column of the form w=(a,O ,..., O)T, a#0 [(O,O,..., O,CX)~,a#O]. Then (i) A does not have any purely imaginary eigenvalues. (ii) In(A) = In( - H). (iii) If in particular H is a positive diagonal matrix (that is, a diagonal matrix having positive elements on its main diagonal), then A is D-stable. Proof. First, suppose that A = (a,) is a upper Hessenberg matrix with non-zero subdiagonal, that is, aii= 0

for

i>j+2

and ai+,,i#O, i=l,2 ,..., n-l. Then the matrix (w,Aw,A2w, ,..., A”-‘w) is an n X n upper triangular matrix with the elements a,a,,a,a,,a,,a,. .., a,+,,.. .a,,- 1a along its main diagonal. Since a #0 and a, + l,i #O, i = 1,2 , . . . ,n - 1, it follows that this matrix is non-singular. The case when A is a lower Hessenberg matrix is similar. In this case, the is an nXn matrix of the form matrix (w,Aw,A2w ,...,A”-‘to) c

. . .

0

0

n-l

...

0

a

II

ai,i

+1

i-l

n-l

...

0

0

. . .

a II

x

ai,i +1

i=2

. . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

. . . . . .

n-1

0

a

II

ai,i+l

X

X

...

x

...

x

...

X

i=n-1

i a Since a#0

x

and ai,i+l#O, i=1,2,...,n-1,

J

this matrix is also non-singular.

BISWA NATH DATTA

138

Thus, in each case, the n x n2 controllability matrix (W, AW, A’W,...,A”-‘W) has full rank n. (i) and (ii) now follow from Theorem 1. To prove (iii), let D be a positive diagonal matrix; then from (2), we have DAHD+

DHATD=

- DWD,

(3)

that is, (DA)HD+HD(DA)~=

-DUD=

- w1

(4)

(since DH = HD). Since A is a Hessenberg matrix with a non-zero codiagonal, so is DA. Also, W1 is a positive semidefinite matrix having the same structure as W. By (ii) above, therefore, we have In(DA)=In(-HD). Since HD is a positive diagonal matrix, DA is a stable matrix.

4.

D-STABLE

MATRICES

A matrix 0 S= i .Yo?.

1

0

1

0

0

. . P. . .I6 . :Ys. ..

...

**. 0 ... 0 . .6 . . . i.

3

0

0

-s,

-s,

is called a Schwarz matrix. It was shown by Schwarz [13] that every nonderogatory matrix A can be transformed into a matrix of the form S by similarity. If S,#O, i=2,3,. . ., n, it is easy to see [S] that there exists a diagonal matrix D

(

Sl

D=dgS,S,...S,’

Sl s,s,...s,_,““’

3s

S,’ l

such that SD+DST=-dg(O,O

,..., 0,2S,2)=-W,

where W is a positive semi-definite matrix.

)

139

STABILITY AND D-STABILITY

By Theorem 2(n), In(S)=In(-D)=In(dg(-S,,-SiS,,...,-SiSs*..S,,)). If S is a stable matrix. then In(S)=(O,n,O)

and

S,>O,

i=1,2 ,..., n

(see also [S]), and th ere fore D is positive diagonal. Also S is clearly a lower Hessenberg matrix with unit superdiagonal. By Theorem 2(E), S is therefore D-stable. Suppose b,, . . . , b,, non-zero. Then b, A=

b,

0

...

0

-b, 0 b3 ... 0 . . . . . . . . . . . . . . . *. . ... ... 0 0 be . . . ... 0 -b,, 0

is a nonderogatory matrix, called a Routh matrix. Let now Ai=A(1,2

,..., ill,2 ,..., i)

denote the leading principal minor of A of order i. Then it can be verified [7j that there exists an &-order

scalar matrix

such that AiDi + DiAiT= Wi =dg(2b;,O,.

. .,O),

that is, A(-Di)+(-Di)AiT=

- Wi=-dg(2b;,O,...,O).

When A is a stable matrix, obviously trace(A) = b,
140

BISWA NATH DA’ITA

P=

***.*

0 1 . .

1 0 . .

;

...

b

b

0

a**

0

0

0 0 . .

0 0 . .

b 1

be a permutation matrix of order 12. Then it is interesting to note that the matrix ...

...

- S, . .l. . . .O. . . . . . . :*:. ... ... ... ...

. 1.:. 1

LS, S’=PSP=

...

_S,

...

0

...

...

0

... :::

0 1

... ’ ,

*** ‘L S,’ 0

is totally stable if S is a stable matrix. For if S/ is the leading principal minor of order i, then the i x i diagonal matrix

is such that S’Di’+D;(Si’)T=

-dg(2S,2,0,...,0)

is negative semidefinite. Clearly, when S is a stable matrix, D/ is positive diagonal (since Si > 0, i = 1,2,. . . , n), and therefore S/ is D-stable. S’ is thus totally stable. Z am grateful to Professor David Carlson of Oregon State University for his interest in this work and his useful comments. FUTEFUZNCES 1 2

K. J. Arrow and M. McManus, A note on dynamic stability, Ecunometrica 26 (1958), 448-454. S. Bamett and C. Storey, Matrix Methods in Stability Theory, Nelson, London,

1970.

STABILITY AND D-STABILITY 3 4 5 6 7 8 9 10 11 12 13 14 15 16

141

D. Carbon, A new criterion for H-stability of complex matrices, Linear Algebra 1 (1968), 59-64. D. Carlson and H. Schneider, Inertia theorems for matrices: the semi-definite case, I. Math. Anal. Appl. 6 (KM%), 430-446. C. T. Chen, A generalization of the inertia theorem, SZAM I. Appl. Math. 25 (1973), 158-161. B. N. Datta, An inertia theorem for the Schwarz matrix, IEEE ‘Zkms. Autm. Control AC-20 (1975), 274. B. N. Datta, On the similarity between a nonderogatory matrix and its Routhcanonical form”, IEEE Trans. Autorn. Control AC-26 (1975), 273-274. A. C. E&oven and K. J, Arrow, A theorem on expectations and the stability of equilibrium, Eccmmw trica 24 (1956), 288-293. C. R. Johnson, D-stability and real and complex quadratic forms, Linear Algebra Appl. 9 (1974), 89-94. C. Il. Johnson, Sufficient conditions for D-stability, J. Econ. Theory 9 (1974), 53-62. A. Ostrowski and H. Schneider, Some theorems on the inertia of general matrices, J. Math. Anal. Appl. 4 (1962), 72-84. J. P. Quirk and R. Ruppert, Qualitative economics and the stability of equilibrium, Reu. Ewn. Stud. 32 (1965), 331326. H. R. Schwarz, Ein Verfahren zur StabiIit&sfrage bei Matrizen-Eigenwertprob lemen, 2. Angew. Math. Whys. 7 (1956), 473500. 0. Tam&y, A remark on a theorem of Lyapunov, 1. Math. And. A$. 2 (1961), 105-107. 0. Taussky, Programmation en mathematiques numeriques, Colloq. Int. du Centre Natl. de la Rech. Sci., Paris, Sept. 1966, 75-88. H. K. Wimmer, Inertia theorems for matrices, controllability and linear vibrations, Linear Algebra Appl. 8 (1974), 337-344. Received

14 April 1977; revised 10 May 1977