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BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access February 12, 2016

Matrix rank/inertia formulas for least-squares solutions with statistical applications

Yongge Tian and Bo Jiang
From the journal Special Matrices

Abstract

Least-Squares Solution (LSS) of a linear matrix equation and Ordinary Least-Squares Estimator (OLSE) of unknown parameters in a general linear model are two standard algebraical methods in computational mathematics and regression analysis. Assume that a symmetric quadratic matrix-valued function Φ(Z) = Q − ZPZ0 is given, where Z is taken as the LSS of the linear matrix equation AZ = B. In this paper, we establish a group of formulas for calculating maximum and minimum ranks and inertias of Φ(Z) subject to the LSS of AZ = B, and derive many quadratic matrix equalities and inequalities for LSSs from the rank and inertia formulas. This work is motivated by some inference problems on OLSEs under general linear models, while the results obtained can be applied to characterize many algebraical and statistical properties of the OLSEs.

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Received: 2015-6-11
Accepted: 2016-2-1
Published Online: 2016-2-12

©2016 Yongge Tian and Bo Jiang

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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