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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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