Representations of the Drazin inverse for a class of block matrices (Q1779265)

The paper presents a formula for the Drazin inverse of the block matrix \( F=\left(\begin{smallmatrix} I_{d} & I_{d} \\ E & 0 \end{smallmatrix}\right)\), where \(E\in \mathbb{C} ^{d\times d}\), ind\(( E) =r\). Further, it gives a representation of the Drazin inverse of \(M=\left(\begin{smallmatrix} A & B \\ C & 0_{d,d} \end{smallmatrix}\right),\) where \(A\in \mathbb{C} ^{d\times d}\), \(B\in \mathbb{C} ^{d\times ( n-d) }\), \(C\in \mathbb{C} ^{( n-d) \times d}\) and \(CA^{D}A=C,A^{D}BC=BCA^{D}\) (\(A^{D}\) denotes the Drazin inverse of \(A\)).