An expression of the Drazin inverse of a perturbed matrix

摘要

Given a square matrix A and its perturbation matrix E, a new expression for the Drazin inverse BD of B=A+E is derived if AADB2=(AADB)2 or B2AAD=(BAAD)2. Based on the new expression, a bound of the relative error of BD is developed. Some known results in the literature on the Drazin inverse and the perturbation bound are included by this new formula as special cases. A numerical example is given to compare the upper bounds.