Characterizations of matrices which eigenprojections at zero are equal to a fixed perturbation (Q706729)

If \(\mathcal{R}(A)\) and \(\mathcal{N}(A)\) denote the range and the null space of a matrix \(A\in\mathbb C^{n\times n}\), then \(\text{ ind}(A)\) is the smallest nonnegative integer \(s\) such that \(\mathcal{R}(A^s)\oplus\mathcal{N}(A^s)=\mathbb C^{n\times n}\). The eigenprojection for eigenvalue 0 is the unique matrix \(A^\pi\) such that \(\mathcal{R}(A^\pi)=\mathcal{N}(A^s)\) and \(\mathcal{N}(A^\pi)=\mathcal{R}(A^s)\). In this paper, characterizations are given for matrices \(B\) satisfying \(B^\pi=A^\pi+S\) with prescribed \(A\) and \(S\). This is used to derive upper bounds for \(\| B^D\| \) and \(\| B^D-A^D\| \) where \(D\) stands for the Drazin inverse and \(\| \cdot\| \) denotes any matrix norm satisfying \(\| AB\| \leq\| A\| \| B\| \).