Complementarity forms of theorems of Lyapunov and Stein

The well known Lyapunov's theorem in matrix theory/ continuous dynamical systems asserts that a (complex) square matrix A is positive stable (that is, all eigenvalues lie in the open right-half plane) if and only if there exists a positive deenite matrix X such that AX+XA is positive deenite. 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 semideenite matrix X such that AX+XA +Q is positive semideenite and XAX+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 semideenite matrix X such that X ? AXA + Q is positive semideenite and XX ? 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 deenite matrix X such that X ? AXA is positive deenite.