Perturbation analysis of the matrix equation X - \sum_{i=1}^m A_i^* X^{p_i} A_i = Q

TL;DR

This paper investigates the nonlinear matrix equation involving positive definite solutions, providing conditions for existence, perturbation bounds, backward error analysis, and condition numbers, supported by numerical examples.

Contribution

It offers new necessary and sufficient conditions for solutions, perturbation bounds, and explicit condition number formulas for the matrix equation with positive exponents.

Findings

Derived conditions for positive definite solutions.

Established perturbation bounds for solutions.

Provided explicit condition number expressions.

Abstract

Consider the nonlinear matrix equation X-sum_{i=1}^{m}A_{i}^{*}X^{p_{i}}A_{i}=Q with p_{i}>0. Sufficient and necessary conditions for the existence of positive definite solutions to the equation with p_{i}>0 are derived. Two perturbation bounds for the unique solution to the equation with 0<p_{i}<1 are evaluated. The backward error of an approximate solution for the unique solution to the equation with 0<p_{i}<1 is given. Explicit expressions of the condition number for the equation with 0<p_{i}<1 are obtained. The theoretical results are illustrated by numerical examples.