Explicit formulae for the Drazin inverse of anti-triangular block matrices

Abstract

In this paper, we analyze the index relation of anti-triangular block matrices and their entries to separately obtain new and strict expressions for the Drazin inverse of them under certain circumstances. As applications, we utilize the relationship between the anti-triangular block matrix and a $2\times2$ block matrix to establish several formulae. Our results generalize and unify a series of results in the literature.