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An improvement on perturbation bounds for the Drazin inverse
Author(s) -
Wei Yimin,
Li Xiezhang
Publication year - 2003
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.336
Subject(s) - drazin inverse , mathematics , inverse , perturbation (astronomy) , combinatorics , upper and lower bounds , rank (graph theory) , square matrix , pure mathematics , mathematical analysis , symmetric matrix , geometry , eigenvalues and eigenvectors , physics , quantum mechanics
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.