On the Hermitian positive definite solutions of nonlinear matrix equation [formula omitted

Abstract

In this paper, the Hermitian positive definite solutions of nonlinear matrix equation Xs+∑i=1mAi∗X-tiAi=Q are considered, where Q is a Hermitian positive definite matrix, Ai are nonsingular complex matrices, s,m are positive numbers, and 0<ti≤1,i=1,…,m. Necessary and sufficient conditions for the existence of Hermitian positive definite solutions are derived. A sufficient condition for the existence of a unique Hermitian positive definite solution is given. In addition, some necessary conditions and sufficient conditions for the existence of Hermitian positive definite solutions are presented. Finally, an iterative method is proposed to compute the maximal Hermitian positive definite solution, and some numerical examples are given to show the efficiency of the proposed iterative method.