On relations among solutions of the Hermitian matrix equation $AXA^{*} = B$ and its three small equations

Abstract

Assume that the linear matrix equation $AXA^{*} = B = B^{*}$ has a Hermitian solution and is partitioned as $\left[\!\! \begin{array}{c} A_1 \\ A_2 \end{array} \!\!\right]\! X[\, A^{*}_1, \, A^{*}_2 \,] = \left[\!\! \begin{array}{cc} B_{11} \,, B_{12} \\ B_{21}^{*} \,, B_{22} \end{array}\right]$. We study in this paper relations among the Hermitian solutions of the equation and the three small-size matrix equations $A_1X_1A^{*}_1 = B_{11}$, $A_1X_2A^{*}_2 = B_{12}$ and $A_2X_3A^{*}_2 = B_{22}$. In particular, we establish closed-form formulas for calculating the maximal and minimal ranks and inertias of $X-X_1-X_2-X_2^{*}-X_3$, and use the formulas to derive necessary and sufficient conditions for the Hermitian matrix equality $X = X_1+X_2+X_2^{*}+X_3$ to hold and Hermitian matrix inequalities $X \geqslant(\leqslant, \cdots)\, X_1+X_2+X_2^{*}+X_3$ to hold in the Löwner partial ordering.