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

摘要

Assume that a quadratic matrix-valued function is given and let be the set of all least-squares solutions of the linear matrix equation . In this paper, we first establish explicit formulas for calculating the maximum and minimum ranks and inertias of subject to , and then derive from the formulas the analytic solutions of the two optimization problems and subject to in the Lowner 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.