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.
The Forward Order Law for Least Squareg-Inverse of Multiple Matrix Products
