A note on the representations for the Drazin inverse of 2x2 block matrices

In 1979, Campbell and Meyer proposed the problem of finding a formula for the Drazin inverse of a 2 x 2 matrix M = [GRAPHICS] in terms of its various blocks, where the blocks A and D are required to be square matrices. Special cases of the problems have been studied. In particular, a representation of the Drazin inverse of M, denoted by M-D, has recently been obtained under the assumptions that C(I - AA(D))B = O and A(I - AA(D))B = O together with the condition that the generalized Schur complement D - CA(D)B be either nonsingular or zero. We derive an alternative representation for M-D under the same assumptions, but with the condition on the Schur complement in the hypothesis replaced by the condition that R(CAA(D)) subset of N(B) boolean AND N(D), where R(.) and N(.) are the range and null space of a matrix. (C) 2007 Elsevier Inc. All rights reserved.