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

Abstract The well-known linear matrix equation A X = 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 − X P 1 X ′ and Q 2 − X ′ P 2 X subject to the restriction trace [ ( A X − B ) ′ W ( A X − 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 A X = B , including necessary and sufficient conditions for weighted least-squares solutions of A X = B to satisfy the quadratic symmetric matrix equalities X P 1 X ′ = Q 1 an X ′ P 2 X = Q 2 , respectively, and necessary and sufficient conditions for the quadratic matrix inequalities XP 1 X ′≻ Q 1  (≽ Q 1 , ≺ Q 1 , ≼ Q 1 ) and X ′ P 2 X ≻ 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 − X P 1 X ′ and Q 2 − X ′ P 2 X subject to weighted least-squares solutions of A X = B .