Papers on Least Squares, Projections, and Generalized Inverses

G.W. Stewart ◽  
2010 ◽  
pp. 340-390
Author(s):  
Misha E. Kilmer ◽  
Dianne P. O’Leary
Keyword(s):  
Least Squares ◽  
Generalized Inverses

scholarly journals A Family of 512 Reverse Order Laws for Generalized Inverses of Two Matrix Product: A Review

2020 ◽  
Author(s):  
Yongge Tian
Keyword(s):  
Least Squares ◽  
Mixed Model ◽  
Linear Mixed Model ◽  
Generalized Inverse ◽  
Sufficient Conditions ◽  
Generalized Inverses ◽  
Matrix Analysis ◽  
Reverse Order ◽  
Matrix Product ◽  
The Matrix

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 $(AB)^{(i,\ldots,j)} = B^{(s_2,\ldots,t_2)}A^{(s_1,\ldots,t_1)}$, where $(\cdot)^{(i,\ldots,j)}$ denotes an $\{i,\ldots, j\}$-generalized inverse of matrix. Because $\{i,\ldots, j\}$-generalized inverse of a singular matrix is unique, the relationships between both sides of the reverse order law can be divided into four situations for consideration. This paper provides a thorough coverage of the reverse order laws for $\{i,\ldots, j\}$-generalized inverses of $AB$, from the development of background and preliminary tools to the collection of miscellaneous formulas and facts on the reverse order laws in one place with cogent introduction and references for further study. We begin with the introduction of a linear mixed model $y = AB\beta + A\gamma + \epsilon$ and the presentation of two least-squares methodologies to estimate the fixed parameter vector $\beta$ in the model, and the description of connections between the two types of least-squares estimators and the reverse order laws for generalized inverses of $AB$. 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 $B^{(s_2,\ldots,t_2)}A^{(s_1,\ldots,t_1)}$, as well as necessary and sufficient conditions for $B^{(s_2,\ldots,t_2)}A^{(s_1,\ldots,t_1)}$ to be invariant with respect to the choice of $B^{(s_2,\ldots,t_2)}A^{(s_1,\ldots,t_1)}$. We then present a unified approach to the 512 matrix set inclusion problems associated with the above reverse order laws for the eight commonly-used types of generalized inverses of $A$, $B$, and $AB$ through use of the definitions of generalized inverses, the block matrix method (BMM), the matrix rank method (MRM), the matrix equation method (MEM), and various algebraic calculations of matrices.

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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