Equalities and inequalities for ranks of products of generalized inverses of two matrices and their applications

Abstract

A complex matrix X is called an {i, … , j} -inverse of the complex matrix A, denoted by A(i,…,j), if it satisfies the ith, …, jth equations of the four matrix equations (i) AXA= A, (ii) XAX= X, (iii) (AX) ∗= AX, (iv) (XA) ∗= XA. The eight frequently used generalized inverses of A are A†, A(1 , 3 , 4), A(1 , 2 , 4), A(1 , 2 , 3), A(1 , 4), A(1 , 3), A(1 , 2), and A(1). The {i, … , j} -inverse of a matrix is not necessarily unique and their general expressions can be written as certain linear or quadratic matrix-valued functions that involve one or more variable matrices. Let A and B be two complex matrices such that the product AB is defined, and let A(i,…,j) and B(i,…,j) be the {i, … , j} -inverses of A and B, respectively. A prominent problem in the theory of generalized inverses is concerned with the reverse-order law (AB) (i,…,j)= B(i,…,j)A(i,…,j). Because the reverse-order products B(i,…,j)A(i,…,j) are usually not unique and can be written as linear or nonlinear matrix-valued functions with one or more variable matrices, the reverse-order laws are in fact linear or nonlinear matrix equations with multiple variable matrices. Thus, it is a tremendous and challenging work to establish necessary and sufficient conditions for all these reverse-order laws to hold. In order to make sufficient preparations in characterizing the reverse-order laws, we study in this paper the algebraic performances of the products B(i,…,j)A(i,…,j). We first establish 126 analytical formulas for calculating the global maximum and minimum ranks of B(i,…,j)A(i,…,j) for the eight frequently used {i, … , j} -inverses of matrices A(i,…,j) and B(i,…,j), and then use the rank formulas to characterize a variety of algebraic properties of these matrix products.