Kronecker product representation for the solution of the general linear matrix equation

1980 ◽  
Vol 25 (3) ◽  
pp. 563-564 ◽  
Author(s):  
R. Weidner ◽  
R. Mulholland
Keyword(s):  
Matrix Equation ◽  
Kronecker Product ◽  
General Linear ◽  
Linear Matrix ◽  
Product Representation ◽  
Linear Matrix Equation

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

2016 ◽  
Vol 4 (1) ◽  
Author(s):  
Yongge Tian ◽  
Bo Jiang
Keyword(s):  
Least Squares ◽  
Matrix Equation ◽  
Linear Models ◽  
Ordinary Least Squares ◽  
General Linear ◽  
Linear Matrix ◽  
Unknown Parameters ◽  
Computational Mathematics ◽  
Linear Matrix Equation ◽  
Quadratic Matrix

AbstractLeast-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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