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

Yongge Tian; Bo Jiang

Special Matrices (2016)

  • Volume: 4, Issue: 1, page 130-140
  • ISSN: 2300-7451

Abstract

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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.

How to cite

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Yongge Tian, and Bo Jiang. "Matrix rank/inertia formulas for least-squares solutions with statistical applications." Special Matrices 4.1 (2016): 130-140. <http://eudml.org/doc/276723>.

@article{YonggeTian2016,
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.},
author = {Yongge Tian, Bo Jiang},
journal = {Special Matrices},
keywords = {Linear model; matrix equation; LSS; OLSE; quadratic matrix-valued function; rank, inertia; linear model; rank; inertia; least squares solution; ordinary least squares estimator; inference problem},
language = {eng},
number = {1},
pages = {130-140},
title = {Matrix rank/inertia formulas for least-squares solutions with statistical applications},
url = {http://eudml.org/doc/276723},
volume = {4},
year = {2016},
}

TY - JOUR
AU - Yongge Tian
AU - Bo Jiang
TI - Matrix rank/inertia formulas for least-squares solutions with statistical applications
JO - Special Matrices
PY - 2016
VL - 4
IS - 1
SP - 130
EP - 140
AB - 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.
LA - eng
KW - Linear model; matrix equation; LSS; OLSE; quadratic matrix-valued function; rank, inertia; linear model; rank; inertia; least squares solution; ordinary least squares estimator; inference problem
UR - http://eudml.org/doc/276723
ER -

References

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