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In this paper the nonlinear matrix equation X-A^{*}X^{-p}A=Q with p>0 isinvestigated. We consider two cases of this equation: the case p>1 and the case0<p<1. In the case p>1, a new sufficient condition for the existence of aunique positive definite solution for the matrix equation is obtained. Aperturbation estimate for the positive definite solution is derived. Explicitexpressions of the condition number for the positive definite solution aregiven. In the case 0<p<1, a new sharper perturbation bound for the uniquepositive definite solution is evaluated. A new backward error of an approximatesolution to the unique positive definite solution is obtained. The theoreticalresults are illustrated by numerical examples.

Solutions and improved perturbation analysis for the matrix equation X-A^{*}X^{-p}A=Q (p>0)