An expression of the Drazin inverse of a perturbed matrix

Given a square matrix A and its perturbation matrix E, a new expression for the Drazin inverse B^D of B=A+E is derived if AA^DB^2=(AA^DB)^2 or B^2AA^D=(BAA^D)^2. Based on the new expression, a bound of the relative error of B^D 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.

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