On deriving the Drazin inverse of a modified matrix

This paper addresses the problem of deriving formulas for the Drazin inverse of a modified matrix. First we focus on obtaining formulas of Sherman-Morrison-Woodbury type for singular matrices, dealing with the Drazin inverse. Denote the Drazin inverse of a square complex matrix A by AD. We provide some expressions of (A CDDB)D in terms of the Drazin inverse of the original matrix A and of its generalized Schur complement Z = D - BA(D)C, where Z is not assumed invertible. In addition, we study some representations of the Drazin inverse of a modified matrix, by using an auxiliary idempotent matrix and some special cases are analyzed. The provided results extend earlier works given in the literature. (C) 2011 Elsevier Inc. All rights reserved.