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We study periodic Lyapunov matrix equations for a general discrete-time linear periodic system $ B_px_p-A_px_{p-1}=f_p $ B p x p − A p x p − 1 = f p , where the matrix coefficients $ B_p $ B p and $ A_p $ A p can be singular. The block coefficients of the inverse operator of the system are referred to as the Green matrices. We derive new decay estimates of the Green matrices in terms of the spectral norms of special solutions to the periodic Lyapunov matrix equations. The study is based on the periodic Schur decomposition of matrices. [ABSTRACT FROM AUTHOR]

SUMMARY A fast algorithm for solving systems of linear equations with banded Toeplitz matrices is studied. An important step in the algorithm is a novel method for the spectral factorization of the generating function associated with the Toeplitz matrix. The spectral factorization is extracted from the right deflating subspaces corresponding to the eigenvalues inside and outside the open unit disk of a companion matrix pencil constructed from the coefficients of the generating function. The factorization is followed by the Woodbury inversion formula and solution of several banded triangular systems. Stability of the algorithm is discussed and its performance is demonstrated by numerical experiments. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY The strong stability of a symplectic matrix is investigated from algorithmic and numerical viewpoints using a theory developed by S.K. Godunov. This theory is based on a different formulation of the Krein-Gelfand-Lidskii characterization of strong stability, better suited for numerical calculations. Copyright © 2011 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]