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 Q(1) - XP1X' and Q(2) - XP2X' subject to the restriction trace[ (AX - B)'W(AX -B)] = min, where both P-i and Q(i) 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' - Q(1) an X'P2X - Q(2), respectively, and necessary and sufficient conditions for the quadratic matrix inequalities XP1X'>Q(1)(>= Q(1)<Q(2), <= Q(2)) and X'/P2X>Q(2)(>= Q(2),<Q(2), <= Q(2)) in the Lowner partial ordering to hold, respectively. In addition, we give closed-form solutions to four LOwner partial ordering optimization problems on Q(1) - XP1X' and Q(2) - X'P2X subject to weighted least-squares solutions of AX = B. (C) 2017 Elsevier Inc. All rights reserved.