Notes on the Hermitian Positive Definite Solutions of a Matrix Equation

Abstract

The nonlinear matrix equation, $X-{\sum }_{i=1}^m{A}_i^{*}{X}^{{\delta }_i}{A}_i=\mathrm{Q,}$ with is investigated. A fixed point theorem in partially ordered sets is proved. And then, by means of this fixed point theorem, the existence of a unique Hermitian positive definite solution for the matrix equation is derived. Some properties of the unique Hermitian positive definite solution are obtained. A residual bound of an approximate solution to the equation is evaluated. The theoretical results are illustrated by numerical examples.