Spectral decompositions for the solutions of Sylvester, Lyapunov, and Krein equations

Abstract

Spectral decompositions for the solutions of Lyapunov equation obtained earlier are generalized to a more general class of solutions of Krein matrix equations including as a special case the standard Sylvester equation. Eigen parts of these decompositions are calculated using residues of matrix resolvents and their derivatives. In particular, spectral decompositions for the solutions of algebraic and discrete Lyapunov equations are obtained in a more general formulation. The practical significance of the obtained spectral expansions is that they allow one to characterize the contribution of individual eigen-components or their pairwise combinations into the asymptotic dynamics of the system perturbation energy.