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.