# Perturbation bounds for the matrix equation $X + A^* \widehat{X}^{-1} A = Q$

## TL;DR

This paper derives perturbation bounds for the maximal positive definite solution of a specific matrix equation involving block matrices, with modifications for certain norm conditions, supported by numerical examples.

## Contribution

It provides new perturbation bounds for solutions of a complex matrix equation, including cases where the norm condition is not met.

## Key findings

- Perturbation bounds for the maximal positive definite solution are established.
- A modified bound is derived for cases where the norm condition is violated.
- Numerical examples illustrate the theoretical results.

## Abstract

Consider the matrix equation $X+ A^*\widehat{X}^{-1}A=Q$, where $Q$ is an $n \times n$ Hermitian positive definite matrix, $A$ is an $mn\times n$ matrix, and $\widehat{X}$ is the $m\times m$ block diagonal matrix with $X$ on its diagonal. In this paper, a perturbation bound for the maximal positive definite solution $X_L$ is obtained. Moreover, in case of $\|\widehat{X_L^{-1}}A\|\ge 1$ a modification of the main result is derived. The theoretical results are illustrated by numerical examples.