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

TL;DR

This paper investigates the nonlinear matrix equation involving inverse terms, establishing the existence and uniqueness of positive definite solutions, deriving perturbation bounds, backward error, and condition numbers, supported by numerical examples.

Contribution

It provides new theoretical results on existence, uniqueness, perturbation bounds, and condition numbers for the matrix equation without restrictions on A_i.

Findings

Existence and uniqueness of positive definite solution established.

Perturbation bounds for the solution derived.

Explicit condition number expressions obtained.

Abstract

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 the equation without any restriction on A_{i}. Three perturbation bounds for the unique solution to the equation are evaluated. A backward error of an approximate solution for the unique solution to the equation is derived. Explicit expressions of the condition number for the unique solution to the equation are obtained. The theoretical results are illustrated by numerical examples.