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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 $$A X= B$$.
##### MSC:
 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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