Improved Results on the Drazin Inverse of a $2\times2$ Block Matrix in Terms of Banachiewicz-Schur Forms

Xifu Liu, Hu Yang

In this paper, we derive a representation for the Drazin inverse of a block matrix $M=\left(\matrix{ A&B\cr C & D}\right)$ under the assumptions $AA^{\pi}B=0$, $CA^{\pi}B=0$, $AA^DBSS^{pi}=0$, $SS^DCWAA^D(AW)^{pi}=0$, and $(AW)^{pi}AA^DBSS^D=0$, where $S=D-CA^DB$ is the generalized Schur complement. And the representation can be regarded as an unified form of $M^D$ because it covers the case either $S$ is nonsingular or zero. Moreover, some alternative representations for the Drazin inverse are presented. Several situations are analyzed and recent results are generalized.