Computational Statistics

, Volume 32, Issue 4, pp 1645–1663 | Cite as

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

  • Yongge Tian
  • Bo Jiang
Original Paper


Assume that a quadratic matrix-valued function \(\psi (X) = Q - X^{\prime }PX\) is given and let \(\mathcal{S} = \left\{ X\in {\mathbb R}^{n \times m} \, | \, \mathrm{trace}[\,(AX - B)^{\prime }(AX - B)\,] = \min \right\} \) 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.


Quadratic matrix-valued function Rank Inertia Löwner partial ordering Linear model 



The authors are grateful to anonymous referees for their helpful comments and suggestions on an earlier version of this paper. This work was supported by the National Natural Science Foundation of China (Grant No. 11271384).


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.China Economics and Management AcademyCentral University of Finance and EconomicsBeijingChina
  2. 2.College of Mathematics and Information ScienceShandong Institute of Business and TechnologyYantaiChina

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