How to establish exact formulas for calculating the max–min ranks of products of two matrices and their generalized inverses

Abstract

This paper investigates the problem of establishing exact formulas for calculating the maximum and minimum ranks of matrix expressions or matrix equalities that involve two matrices and their generalized inverses of matrices. For a pair of matrices A and B such that the product AB is defined, two fundamental matrix expressions or matrix equalities composed by their generalized inverses are given by and , where and are the -inverses of A and B, respectively. Recently, the present author established a group of exact formulas for calculating the max–min ranks of the products , and derived many fundamental algebraic properties of from the rank formulas. As a continuation, we study in this paper the ranks of the multiple matrix products , , , , , as well as the matrix differences , , , . All these kinds of problems are in fact to characterize algebraic properties of certain matrix-valued functions and we need to utilize a variety of known matrix rank formulas during this tedious work.