Representations for the Drazin Inverse of a 2 x 2 Block Matrix

Abstract

Two representations for the Drazin inverse of a $2\times2$ block matrix $M=[{A \atop C}\;{B \atop D}]$, where A and D are square matrices, in terms of the Drazin inverses of A and D have been recently developed under the assumptions that $C(I-AA^{D})=0$ and $(I-AA^{D})B=0$, and that the generalized Schur complement $D-CA^{D}B$ is either nonsingular or zero. These two representations of $M^{D}$ are extended to the case where $C(I-AA^{D})=0$ and $(I-AA^{D})B=0$ are substituted with $C(I-AA^{D})B=0$ and $A(I-AA^{D})B=0$. Moreover, upper bounds for the index of M are studied. Numerical examples are given to illustrate the new results.