Solutions and Improved Perturbation Analysis for the Matrix Equation X - A * X - p A = Q ( p > 0 )

Abstract

The nonlinear matrix equation $X-A^{\mathrm{*}}{X}^{-p}A=Q$ with $p>\mathrm{0}$ is investigated. We consider two cases of this equation: the case $p\ge \mathrm{1}$ and the case In the case $p\ge \mathrm{1}$, a new sufficient condition for the existence of a unique positive definite solution for the matrix equation is obtained. A perturbation estimate for the positive definite solution is derived. Explicit expressions of the condition number for the positive definite solution are given. In the case , a new sharper perturbation bound for the unique positive definite solution is derived. A new backward error of an approximate solution to the unique positive definite solution is obtained. The theoretical results are illustrated by numerical examples.