Further Expressions on the Drazin Inverse for Block Matrix

Abstract

This article addresses the problem of developing new expressions for the Drazin inverse of complex block matrix $$M=\left( \begin{array}{cc} A &{} B \\ C &{} D \\ \end{array} \right) \in {\mathbb {C}}^{n\times n}$$ (where A and D are square matrices but not necessarily of the same size) in terms of the Drazin inverse of matrix A and of its generalized Schur complement $$S=D-CA^DB$$ which is not necessarily invertible. This formula is the extension of the well-known Banachiewicz inversion formula of complex block matrix M. In addition, we provide representation for the Drazin inverse of complex block matrix M without any restriction on the generalized Schur complement S and under different conditions than those used in some current literature on this subject. Finally, several illustrative numerical examples are considered to demonstrate our results.