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

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.