Closed-form formulas for calculating the max–min ranks of a triple matrix product composed by generalized inverses

In matrix theory and its applications, people often meet with various matrix expressions or matrix equalities that involve inverses of nonsingular matrices or generalized inverses of singular matrices. One of the fundamental research problems about these matrix expressions is to determine their singularity and nonsingularity, or alternatively, to determine upper and lower bounds of the ranks of the matrix expressions. Matrix rank formulas were introduced in the development of generalized inverse theory in 1970s, and many fundamental closed-form formulas for calculating ranks of matrices and their generalized inverses were established. In this paper, we consider the problem of determining the maximum and minimum ranks of a triple matrix product \(P^{(i,\ldots ,j)}NQ^{(i,\ldots ,j)}\), where P, Q, and N are given matrices, \(P^{(i,\ldots ,j)}\) and \(Q^{(i,\ldots ,j)}\) are \(\{i,\ldots , j\}\)-generalized inverses of P and Q, respectively. We first rewrite the product as certain linear or nonlinear matrix-valued function with one or more variable matrices. We then establish a family of closed-form formulas for calculating the maximum and minimum ranks of \(P^{(i,\ldots ,j)}NQ^{(i,\ldots ,j)}\) with respect to \(\{i,\ldots , j\}\)-inverses of P and Q, respectively. Some applications of these max–min rank formulas are also discussed.