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Consider the nonlinear matrix equation X-sum_{i=1}^{m}A_{i}^{*}X^{-1}A_{i}=Q.This paper shows that there exists a unique positive definite solution to theequation without any restriction on A_{i}. Three perturbation bounds for theunique solution to the equation are evaluated. A backward error of anapproximate solution for the unique solution to the equation is derived.Explicit expressions of the condition number for the unique solution to theequation are obtained. The theoretical results are illustrated by numericalexamples.

Solutions and perturbation analysis of the matrix equation X - \sum_{i=1}^m A_i^* X^{-1} A_i = Q