an:06835521
Zbl 1417.15018
Tian, Yongge; Jiang, Bo
Quadratic properties of least-squares solutions of linear matrix equations with statistical applications
EN
Comput. Stat. 32, No. 4, 1645-1663 (2017).
00371537
2017
j
15A24 15B57 62J05 62H12
quadratic matrix-valued function; rank; inertia; L??wner partial ordering; linear model
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 L??wner 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.