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

Version 2 : Received: 29 April 2020 / Approved: 30 April 2020 / Online: 30 April 2020 (05:28:55 CEST)

How to cite:
Tian, Y. Linear Statistical Models, Least-Squares Estimators, and Classication Analysis to Reverse-Order Laws for Generalized Inverses of Matrix Products. *Preprints* **2018**, 2018120342 (doi: 10.20944/preprints201812.0342.v1).
Tian, Y. Linear Statistical Models, Least-Squares Estimators, and Classication Analysis to Reverse-Order Laws for Generalized Inverses of Matrix Products. Preprints 2018, 2018120342 (doi: 10.20944/preprints201812.0342.v1).

## Abstract

*AB*is (

*AB*)

^{(i,...,j)}=

*B*

^{(i,...,j}

^{)}

*A*, where (·)

^{(i,...,j)}*denotes an {*

^{i,...,j}*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 {(·)

^{(}*} denotes the collection of all {*

^{i,...,j)}*i,...,j*}-generalized inverse of a matrix.

## Subject Areas

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