Equalities and inequalities for Hermitian solutions and Hermitian definite solutions of the two matrix equations $AX=B$ and $AX{A}^{\ast }=B$

Abstract

This paper studies algebraic properties of Hermitian solutions and Hermitian definite solutions of the two types of matrix equation $AX=B$ and $AX{A}^{\ast }=B$. 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 $AX{A}^{\ast }=B$, 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.

22 pages