Preprint Review Version 2 Preserved in Portico This version is not peer-reviewed

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

Version 1 : Received: 27 December 2018 / Approved: 28 December 2018 / Online: 28 December 2018 (08:19:32 CET)
Version 2 : Received: 29 April 2020 / Approved: 30 April 2020 / Online: 30 April 2020 (05:28:55 CEST)

How to cite: Tian, Y. A Family of 512 Reverse Order Laws for Generalized Inverses of Two Matrix Product: A Review. Preprints 2018, 2018120342. https://doi.org/10.20944/preprints201812.0342.v2. Tian, Y. A Family of 512 Reverse Order Laws for Generalized Inverses of Two Matrix Product: A Review. Preprints 2018, 2018120342. https://doi.org/10.20944/preprints201812.0342.v2.

Abstract

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.

Keywords

matrix product; orthogonal projector; generalized inverse; reverse order law; BMM; MEM; MRM

Subject

MATHEMATICS & COMPUTER SCIENCE, General Mathematics

Comment 1
Commenter: Yongge Tian
Commenter's Conflict of Interests: Author
Comment: Add new results and facts with corrections.
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