A family of 512 reverse order laws for generalized inverses of a matrix product: A review

Reverse order laws for generalized inverses of matrix products are 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 ${\left(AB\right)}^{(i,\dots ,j)}=B^{(s_2,\dots ,t_2)}{A}^{(s_1,\dots ,t_1)}$ , where ${(\cdot )}^{(i,\dots ,j)}$ denotes a $\{i,\dots ,j\}$ -generalized inverse of matrix. Because $\{i,\dots ,j\}$ -generalized inverse of a singular matrix is not unique, the relationships between both sides of the reverse order law can be divided into four situations for consideration. The aim of this paper is to give an overview of plenty of results concerning reverse order laws for $\{i,\dots ,j\}$ -generalized inverses of the product 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 $y=AB\beta +A\gamma +ϵ$ and the presentation of two least-squares methodologies for estimating the fixed parameter vector β 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 various necessary matrix study tools, including a general theory on linear or nonlinear algebraic 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 $B^{(s_2,\dots ,t_2)}{A}^{(s_1,\dots ,t_1)}$ , as well as necessary and sufficient conditions for $B^{(s_2,\dots ,t_2)}{A}^{(s_1,\dots ,t_1)}$ to be invariant with respect to the choice of the generalized inverses, etc. Subsequently, we 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 by means of the definitions of generalized inverses, the block matrix methodology (BMM), the matrix equation methodology (MEM), and the matrix rank methodology (MRM).