On the Hermitian solutions of the matrix equation X^s + A^*Χ^(-s)A = Q

摘要

In this paper, necessary and sufficient conditions for the existence of the Hermitian solutions of the nonlinear matrix equation Xs+A*X-sA= Q are presented, when A is a nonsingular matrix and s an integer. The formulas for the computation of these solutions are presented. An algebraic method for the computation of the solutions is proposed; the method is based on the algebraic solution of the corresponding discrete time Riccati equation. The exact number of the Hermitian solutions is also derived. The formula for the computation of the maximal solution of the matrix equation Xs-A* X-s A= Q is given as an application of the formulas derived for solving Xs+A* X-s =Q. The results are verified through simulation experiments.