Error bounds for the perturbation of the Drazin inverse under some geometrical conditions

This paper studies the Drazin inverse for perturbed matrices. For that, given a square matrix A, we consider and characterize the class of matrices B with index s   such that R(BAD)∩N(AD)={0},N(ADB)∩R(AD)={0}R(BAD)∩N(AD)={0},N(ADB)∩R(AD)={0}, and R(Bs)=R(BAD)R(Bs)=R(BAD), where N(A)N(A) and R(A)R(A) denote the null space and the range space of a matrix A  , respectively, and ADAD denote the Drazin inverse of A  . Then, we provide explicit representations for BDBD and BBDBBD, and upper bounds for the relative error ‖BD-AD‖/‖AD‖‖BD-AD‖/‖AD‖ and the error ‖BBD-AAD‖‖BBD-AAD‖. A numerical example illustrates that the obtained bounds are better than others given in the literature.