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Quadratic properties of least-squares solutions of linear matrix equations with statistical applications. (English) Zbl 1417.15018
Summary: Assume that a quadratic matrix-valued function $$\psi (X) = Q - X'PX$$ is given and let $$\mathcal{S} = \left\{ X\in {\mathbb R}^{n \times m} \mid \text{trace}[\,(AX - B)'(AX - B)\,] = \min \right\}$$ be the set of all least-squares solutions of the linear matrix equation $$AX = B$$. In this paper, we first establish explicit formulas for calculating the maximum and minimum ranks and inertias of $$\psi (X)$$ subject to $$X \in {\mathcal S}$$, and then derive from the formulas the analytic solutions of the two optimization problems $$\psi (X) =\max$$ and $$\psi (X)= \min$$ subject to $$X \in \mathcal{S}$$ in the Löwner partial ordering. As applications, we present a variety of results on equalities and inequalities of the ordinary least squares estimators of unknown parameter vectors in general linear models.

##### MSC:
 15A24 Matrix equations and identities 15B57 Hermitian, skew-Hermitian, and related matrices 62J05 Linear regression; mixed models 62H12 Estimation in multivariate analysis
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