Abstract
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 (AB)(i,…,j)=B(s2,…,t2)A(s1,…,t1), where (⋅)(i,…,j) denotes a {i,…,j}-generalized inverse of matrix. Because {i,…,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,…,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=ABβ+Aγ+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(s2,…,t2)A(s1,…,t1), as well as necessary and sufficient conditions for B(s2,…,t2)A(s1,…,t1) 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).
Highlights
We begin with introducing the notation adopted in this paper
We conclude that this expository article studies primarily the relationships between (AB)(i,...,j) and B(s2,...,t2)A(s1,...,t1) for the eight commonly-used types of generalized inverses of A, B, and AB by means of the three conventional yet distinguished methods in matrix theory— the matrix rank method, the block matrix method, and the matrix equation method, which brings together many conclusions on the matrix set inclusions in (1.10) in one organized, comprehensive, and self-contained place and provides a valuable source of reference
The whole work has contributed to our present understanding of many formulas, results, and facts in the theory of generalized inverses of matrices, and it is believed that the contributions in this paper will have a profound impact on the development of matrix equality theory, and many more profound and fruitful investigations can be conducted on various equality problems of matrix-valued functions and generalized inverses of matrix products
Summary
We begin with introducing the notation adopted in this paper. Let Rm×n and Cm×n denote the collections of all m × n real and complex matrices, respectively; AT and A∗ denote the transpose and the conjugate transpose, respectively; r(A) denote the rank of A, i.e., the maximum order of the invertible submatrix of A; R(A) = {Ax | x ∈ Cm×1} and N(A) = {x ∈ Cn×1 | Ax = 0} denote the range and the null space of a matrix A ∈ Cm×n, respectively; Im denote the identity matrix of order m; and [A, B] denote a columnwise partitioned matrix consisting of two submatrices A and B. The corresponding work includes simplifying and deriving various complicated and nuanced matrix expressions and equalities, characterizing various ROLs for generalized inverses of matrix products, and establishing thousands of closed-form formulas for calculating ranks of block matrices, sums and differences of matrices, etc (cf [78, 79, 81, 83, 84, 85, 86, 87, 88, 89, 90, 91, 93, 94, 95, 97, 98, 99, 100, 101, 102, 105, 106, 107, 109]). It is imperative to establish necessary and sufficient conditions for the four matrix equalities in (2.25)–(2.28) to hold in order to interpret and use the four statistical statements in (2.25)–(2.28)
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