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

Abstract

In 1979, Campbell and Meyer proposed the problem of finding a formula for the Drazin inverse of a 2 × 2 matrix $M=\left[\begin{array}{cc}\hfill A\hfill & \hfill B\hfill \\ \hfill C\hfill & \hfill D\hfill \end{array}\right]$ 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^DB 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 $\mathcal{R}({\mathit{CAA}}^D)\subset \mathcal{N}(B)\cap \mathcal{N}(D)$, where $\mathcal{R}(·)$ and $\mathcal{N}(·)$ are the range and null space of a matrix.