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In this paper, necessary and sufficient conditions for the existence of the positive definite solutions of the nonlinear matrix equation X + A*X ! A =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.

On the Positive Definite Solutions of the Matrix Equation XS + A*X-S A = Q