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Rank/inertia approaches to weighted least-squares solutions of linear matrix equations. (English) Zbl 1426.15020
Summary: 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 \(Q_1 - X P_1 X^\prime\) and \(Q_2 - X^\prime P_2 X\) subject to the restriction trace \([(A X - B)^\prime W(A X - B)] = \min \), 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 \(AX=B\), including necessary and sufficient conditions for weighted least-squares solutions of \(A X= B\) to satisfy the quadratic symmetric matrix equalities \(XP_1 X^\prime = Q_1\) an \(X^\prime P_2 X = Q_2\), respectively, and necessary and sufficient conditions for the quadratic matrix inequalities \(XP_1X'\succ Q_1 (\succcurlyeq Q_1, \prec Q_1, \preccurlyeq Q_1)\) and \(X'P_2X\succ Q_2 (\succcurlyeq Q_2, \prec Q_2, \preccurlyeq Q_2)\) in the Löwner partial ordering to hold, respectively. In addition, we give closed-form solutions to four Löwner partial ordering optimization problems on \(Q_1 - X P_1 X^\prime\) and \(Q_2 - X^\prime P_2 X\) subject to weighted least-squares solutions of \(A X= B\).
15A24 Matrix equations and identities
15A09 Theory of matrix inversion and generalized inverses
62H12 Estimation in multivariate analysis
62J05 Linear regression; mixed models
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