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

The well-known Lyapunov's theorem in matrix theory/continuous dynamical sys tems 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 th is paper, we prove a complementarity form of this theorem: A is positive st able if and only if for any Hermitian matrix Q, there exists a positive sem idefinite 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) matr ix 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 -1), we deduce the well-known Stein's theorem in discrete linear dynamical systems: A has all eigenvalues in the open uni t disk if and only if there exists a positive definite matrix X such that X - AXA* is positive definite. (C) 2000 Elsevier Science Inc, All rights res erved.