Abstract
This paper studies algebraic properties of Hermitian solutions and Hermitian definite solutions of the two types of matrix equations 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.
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