Rank/inertia approaches to weighted least-squares solutions of linear matrix equations

The well-known linear matrix equation AX=B is the simplest representative of all linear matrix equations. In this paper, we study quadratic properties of weighted least-squares solutions of this matrix equation. We first establish two groups of closed-form formulas for calculating the global maximum and minimum ranks and inertias of matrices in the two quadratical matrix-valued functions Q1−XP1X′ and Q2−X′P2X subject to the restriction trace[(AX−B)′W(AX−B)]=min, where both Pi and Qi are real symmetric matrices, i=1,2, W is a positive semi-definite matrix, and X′ is the transpose of X. We then use the rank and inertia formulas to characterize quadratic properties of weighted least-squares solutions of AX=B, including necessary and sufficient conditions for weighted least-squares solutions of AX=B to satisfy the quadratic symmetric matrix equalities XP1X′=Q1 an X′P2X=Q2, respectively, and necessary and sufficient conditions for the quadratic matrix inequalities XP1X′≻Q1 (≽Q1, ≺Q1, ≼Q1) and X′P2X≻Q2 (≽Q2, ≺Q2, ≼Q2) in the Löwner partial ordering to hold, respectively. In addition, we give closed-form solutions to four Löwner partial ordering optimization problems on Q1−XP1X′ and Q2−X′P2X subject to weighted least-squares solutions of AX=B.