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A constructive perturbation bound of the Drazin inverse of a square matrix is derived using a technique proposed by G. Stewart and based on perturbation theory for invariant subspaces. This is an improvement of the result published by the authors Wei and Li [Numer. Linear Algebra Appl., 10 (2003), pp. 563-575]. It is a totally new approach to developing perturbation bounds for the Drazin inverse of a matrix. A numerical example which indicates the sharpness of the perturbation bound is presented.

SUMMARYThe Drazin inverse of a square matrix occurs in a number of applications. It is of importance to analyse the perturbation bounds for the Drazin inverse of a matrix. Let B = A + E. Under the assumption of rank(B j ) = rank(A k ), where j and k are the indices of B and A, respectively, upper bounds ofHowever, these upper bounds do not cover the perturbation bounds of the group inverse recently given by the authors as a special case.Moreover, these perturbation bounds for the Drazin inverse are too large to be practical. In this paper, we present sharper uniÿed perturbation bounds for the Drazin inverse, which are the extensions of the recent result in the case of group inverse. It solves the problem posed by Campbell and Meyer in 1975. A numerical example is given to illustrate the sharpness of the new general bounds.

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