ON THE EXISTENCE OF POSITIVE DEFINITE SOLUTIONS OF A NONLINEAR MATRIX EQUATION

Abstract

In this paper the nonlinear matrix equation $X-\sum\limits_{i=1}^{m}A_{i}^*X^{-p_{i}}A_{i}=Q $ with $p_{i}\gt 0$ is investigated. Necessary and sufficient conditions for the existence of Hermitian positive definite solutions are obtained. An effective iterative method to obtain the unique solution is established. A perturbation bound and the backward error of an approximate solution to this solution is evaluated. Moreover, an explicit expression of the condition number for the positive definite solution is given. The theoretical results are illustrated by numerical examples.