An improvement on perturbation bounds for the Drazin inverse
Yimin Wei, Xiezhang Li · Numerical Linear Algebra with Applications · 2003
Abstract The 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 of ∥ B D ‐ A D ∥/∥ A D ∥ and ∥ BB D ‐ AA D ∥/∥ AA D ∥ have been recently studied. However, 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 unified 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. Copyright © 2003 John Wiley & Sons, Ltd.