The Open Applied Mathematics Journal
2011, 5 : 19-25Published online 2011 November 30. DOI: 10.2174/1874114201105010019
Publisher ID: TOAMJ-5-19
On the Positive Definite Solutions of the Matrix Equation XS + A* X− S A = Q
Department of Computer Science and Biomedical Informatics, University of Central Greece, Lamia
35100, Greece.
ABSTRACT
In this paper, necessary and sufficient conditions for the existence of the positive definite solutions of the nonlinear matrix equation Xs + A* X!sA =Q are presented, when A is nonsingular and s an integer, as well as some new properties and bounds for the eigenvalues of matrices related to A,Q are discussed. An algebraic method for the computation of the solutions is proposed, based on the algebraic solution of the corresponding discrete time Riccati equation. The exact number of the positive definite solutions is also computed. Mathematics Subject Classification: 15A60, 15A18, 15A24, 93B40.