Common Hermitian least squares solutions of matrix equations A1XA1∗=B1 and A2XA2∗=B2 subject to inequality restrictions

In this paper, we give some closed-form formulas for calculating maximal and minimal ranks and inertias of P−XP−X with respect to XX, where P∈CHn is given, XX is common Hermitian least squares solutions to matrix equations A1XA1∗=B1 and A2XA2∗=B2. As application, we derive necessary and sufficient conditions for X>P(≥P,