Complementarity forms of theorems of Lyapunov and Stein, and related results

Seetharama Gowda, M. ; Parthasarathy, T. (2000) Complementarity forms of theorems of Lyapunov and Stein, and related results Linear Algebra and its Applications, 320 (1-3). pp. 131-144. ISSN 0024-3795

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Official URL: http://www.sciencedirect.com/science/article/pii/S...

Related URL: http://dx.doi.org/10.1016/S0024-3795(00)00208-1

Abstract

The well-known Lyapunov's theorem in matrix theory / continuous dynamical systems asserts that a (complex) square matrix A is positive stable (i.e., all eigenvalues lie in the open right-half plane) if and only if there exists a positive definite matrix X such that AX+XA is positive definite. In this paper, we prove a complementarity form of this theorem: A is positive stable if and only if for any Hermitian matrix Q, there exists a positive semidefinite matrix X such that AX+XA+Q is positive semidefinite and X[AX+XA+Q]=0. By considering cone complementarity problems corresponding to linear transformations of the form I-S, we show that a (complex) matrix A has all eigenvalues in the open unit disk of the complex plane if and only if for every Hermitian matrix Q, there exists a positive semidefinite matrix X such that X-AXA+Q is positive semidefinite and X[X-AXA+Q]=0. By specializing Q (to -I), we deduce the well known Stein's theorem in discrete linear dynamical systems: A has all eigenvalues in the open unit disk if and only if there exists a positive definite matrix X such that X-AXA is positive definite.

Item Type:Article
Source:Copyright of this article belongs to Elsevier Science.
ID Code:90946
Deposited On:15 May 2012 09:59
Last Modified:15 May 2012 09:59

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