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

TL;DR

This paper investigates a nonlinear matrix equation involving positive powers, providing new conditions for solution existence, perturbation bounds, and explicit condition numbers, supported by numerical examples.

Contribution

The paper introduces new sufficient conditions for solution existence and sharper perturbation bounds for the matrix equation with p>0, extending previous results.

Findings

Established a new sufficient condition for p>1 case.

Derived a sharper perturbation bound for 0<p<1.

Provided explicit expressions for the condition number.

Abstract

In this paper the nonlinear matrix equation X-A^{*}X^{-p}A=Q with p>0 is investigated. We consider two cases of this equation: the case p>1 and the case 0<p<1. In the case p>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 0<p<1, a new sharper perturbation bound for the unique positive definite solution is evaluated. A new backward error of an approximate solution to the unique positive definite solution is obtained. The theoretical results are illustrated by numerical examples.