Equalities and inequalities for Hermitian solutions and Hermitian definite solutions of the two matrix equations $AX = B$ and $AXA^* = B$

TL;DR

This paper investigates algebraic properties, rank, and inertia formulas for Hermitian and Hermitian definite solutions of matrix equations $AX = B$ and $AXA^* = B$, providing new characterizations and inequalities.

Contribution

It introduces new rank and inertia formulas for Hermitian solutions and Hermitian definite solutions of the two matrix equations, enabling detailed analysis and inequalities.

Findings

Derived formulas for maximal and minimal ranks and inertias.

Characterized properties and inequalities of Hermitian solutions.

Provided comprehensive algebraic analysis of the solutions.

Abstract

This paper studies algebraic properties of Hermitian solutions and Hermitian definite solutions of the two types of matrix equation $AX = B$ and $AXA^* = B$. We first establish a variety of rank and inertia formulas for calculating the maximal and minimal ranks and inertias of Hermitian solutions and Hermitian definite solutions of the matrix equations $AX = B$ and $AXA^* = B$, and then use them to characterize many qualities and inequalities for Hermitian solutions and Hermitian definite solutions of the two matrix equations and their variations.