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

Abstract

A complex matrix X is called an $\{i,\ldots, j\}$ -inverse of the complex matrix A, denoted by $A^{(i,\ldots, j)}$ , 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 $A^{\dagger}$ , $A^{(1,3,4)}$ , $A^{(1,2,4)}$ , $A^{(1,2,3)}$ , $A^{(1,4)}$ , $A^{(1,3)}$ , $A^{(1,2)}$ , and $A^{(1)}$ . The $\{i,\ldots, j\}$ -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 $A^{(i,\ldots ,j)}$ and $B^{(i,\ldots,j)}$ be the $\{i,\ldots, j\}$ -inverses of A and B, respectively. A prominent problem in the theory of generalized inverses is concerned with the reverse-order law $(AB)^{(i,\ldots,j)} = B^{(i,\ldots,j)}A^{(i,\ldots,j)}$ . Because the reverse-order products $B^{(i,\ldots,j)}A^{(i,\ldots,j)}$ 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 $B^{(i,\ldots,j)}A^{(i,\ldots,j)}$ . We first establish 126 analytical formulas for calculating the global maximum and minimum ranks of $B^{(i,\ldots,j)}A^{(i,\ldots,j)}$ for the eight frequently used $\{i,\ldots, j\}$ -inverses of matrices $A^{(i,\ldots,j)}$ and $B^{(i,\ldots,j)}$ , and then use the rank formulas to characterize a variety of algebraic properties of these matrix products.