Common Hermitian least squares solutions of matrix equations A1Xa*1 and A2Xa*2 subject to inequality restrictions

In this paper, we give some closed‐form formulas for calculating maximal and minimal ranks and inertias of $P‐X$ with respect to $X$ , where $P\in C_H^n$ is given, $X$ is common Hermitian least squares solutions to matrix equations $A_{1}X{A}_1^{*}=B_1$ and $A_{2}X{A}_2^{*}=B_2$ . As application, we derive necessary and sufficient conditions for $X>P(\ge P,<P,\le P)$ in the Löwner partial ordering. In addition, we give identifying conditions for the existence of definite common Hermitian least squares solutions to matrix equations $A_{1}X{A}_1^{*}=B_1$ and $A_{2}X{A}_2^{*}=B_2$ .