Two groups of mixed reverse order laws for generalized inverses of two and three matrix products

2020 ◽  
Vol 39 (3) ◽  
Author(s):  
Yongge Tian
Keyword(s):  
Generalized Inverses ◽  
Reverse Order ◽  
Matrix Products

scholarly journals The mixed-type reverse order laws for generalized inverses of the product of two matrices

Filomat ◽  
2013 ◽  
Vol 27 (5) ◽  
pp. 937-947
Author(s):  
Zhiping Xiong
Keyword(s):  
Mixed Type ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Generalized Inverses ◽  
Reverse Order ◽  
Matrix Products ◽  
Schur Complements ◽  
Necessary And Sufficient ◽  
The Relationship

The relationship between generalized inverses of AB and the product of generalized inverses of A and B have been studied in this paper. The necessary and sufficient conditions for a number of mixed-type reverse order laws of generalized inverses of two matrix products are derived by using the maximal ranks of the generalized Schur complements.

scholarly journals Linear Statistical Models, Least-Squares Estimators, and Classication Analysis to Reverse-Order Laws for Generalized Inverses of Matrix Products

2018 ◽  
Author(s):  
Yongge Tian
Keyword(s):  
Least Squares ◽  
Linear Mixed Model ◽  
Generalized Inverse ◽  
Sufficient Conditions ◽  
Generalized Inverses ◽  
Matrix Analysis ◽  
Reverse Order ◽  
Matrix Product ◽  
Least Squares Estimators ◽  
Matrix Products

Reverse-order laws for generalized inverses of matrix products is a classic object of study in the theory of generalized inverses. One of the well-known reverse-order laws for a matrix product AB is (AB)(i,...,j) = B(i,...,j)A(i,...,j), where (·)i,...,j denotes an {i,...,j}-generalized inverse of matrix. Because {i,...,j}-generalized inverse of a general matrix is not necessarily unique, the relationships between both sides of the reverse-order law can be divided into four situations for consideration. In this article, we first introduce a linear mixed model y = ABβ + Aγ + ε, present two least-squares  methodologies to estimate the fixed parameter vector in the model, and describe the connections between the two least-squares estimators and the reverse-order laws for generalized inverses of the matrix product AB. We then prepare some valued matrix analysis tools, including a general theory on linear or nonlinear matrix identities, a group of expansion formulas for calculating ranks of block matrices, two groups of explicit formulas for calculating the maximum and minimum ranks of B(i,...,j)A(i,...,j), as well as necessary and sufficient conditions for B(i,...,j)A(i,...,j) to be invariant with respect to the choice of A(i,...,j) and B(i,...,j). We then present a unied approach to the 512 set inclusion problems {(AB)(i,...,j) ⊇ {B(i,...,j)A(i,...,j)}for the eight commonly-used types of generalized inverses of A, B, and AB using the block matrix representation method (BMRM), matrix equation method (MEM), and matrix rank method (MRM), where {(·)(i,...,j)} denotes the collection of all {i,...,j}-generalized inverse of a matrix.

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