Quadratic properties of least-squares solutions of linear matrix equations with statistical applications

Assume that a quadratic matrix-valued function $$\psi (X) = Q - X^{\prime }PX$$ ψ ( X ) = Q - X ′ P X is given and let $$\mathcal{S} = \left\{ X\in {\mathbb R}^{n \times m} \, | \, \mathrm{trace}[\,(AX - B)^{\prime }(AX - B)\,] = \min \right\} $$ S = X ∈ R n × m | trace [ ( A X - B ) ′ ( A X - B ) ] = min be the set of all least-squares solutions of the linear matrix equation $$AX = B$$ A X = B . In this paper, we first establish explicit formulas for calculating the maximum and minimum ranks and inertias of $$\psi (X)$$ ψ ( X ) subject to $$X \in {\mathcal S}$$ X ∈ S , and then derive from the formulas the analytic solutions of the two optimization problems $$\psi (X) =\max $$ ψ ( X ) = max and $$\psi (X)= \min $$ ψ ( X ) = min subject to $$X \in \mathcal{S}$$ X ∈ 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.