A Hermitian least squares solution of the matrix equation AXB=CAXB=C subject to inequality restrictions

This paper gives some closed-form formulas for computing the maximal and minimal ranks and inertias of P−XP−X with respect to XX, where P∈CHn is given, and XX is a Hermitian least squares solution to the matrix equation AXB=CAXB=C. We derive, as applications, necessary and sufficient conditions for X⩾(⩽,>,<)P in the Löwner partial ordering. In addition, we give necessary and sufficient conditions for the existence of a Hermitian positive (negative, nonpositive, nonnegative) definite least squares solution to AXB=CAXB=C.