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.

中文翻译：

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块矩阵Drazin逆的进一步表达式

本文解决了开发复杂块矩阵Drazin逆的新表达式的问题\（M = \ left（\ begin {array} {cc} A＆{} B \\ C＆{} D \\ \ end {array } \右）\在{\ mathbb {C}} ^ {N \ n次} \） （其中，甲和d是正方形矩阵，但是在基质的Drazin逆而言不一定同样大小）甲和其广义舒尔补数\（S = D-CA ^ DB \）不一定是可逆的。该公式是复数块矩阵M的著名Banachiewicz反演公式的扩展。此外，我们提供了复杂块矩阵M的Drazin逆的表示对广义Schur补体S没有任何限制，并且在与当前有关该主题的一些文献所使用的条件不同的条件下。最后，考虑了几个说明性的数值示例来证明我们的结果。