Additive perturbation results for the Drazin inverse (Q1774977)

The paper deals with additive results for the Drazin inverse of complex matrices of size \(n \times n\). Given an \(n \times n\) complex matrix \(A\), let \(A^D\) denote the Drazin inverse of \(A\), \(A^{\pi}\) the eigenprojection of \(A\) corresponding to the eigenvalue 0 and \(ind(A)\) the index of \(A\). By assuming the conditions \(A^DB=0\), \(AB^D=0\) and \(B^{\pi}ABA^{\pi}=0\), the author obtains a formula for the Drazin inverse \((A+B)^D\) as a function of \(A\), \(B\), \(A^D\), \(B^D\), \(A^{\pi}\) and \(B^{\pi}\). The author considers some applications of his results to the perturbation of the Drazin inverse and analyzes a number of special cases. He obtains a perturbation result that generalizes those obtained by \textit{R. E. Hartwig, G. Wang} and \textit{Y. Wei} [ibid. 322, 207--217 (2001; Zbl 0967.15003)].