 
Summary: Perturbation analysis of reduced resolvents and generalized
inverses
Konstantin E. Avrachenkov \Lambda Jean B. Lasserre y
May 1, 2001
Abstract
We investigate analytic perturbations of the reduced resolvent of a ønitedimensional
linear operator (also known as Drazin inverse in the linear algebra literature). Our ap
proach is based on spectral theory of linear operators as well as on a new notion of group
reduced resolvent. It allows to treat regular and singular perturbations in a uniøed frame
work. We provide an algorithm for computing the coeOEcients of the Laurent series of the
perturbed reduced resolvent. In particular, the regular part coeOEcients can be calculated
by simple recursive formulae. Finally, we apply these results to the perturbation analysis
of MoorePenrose generalized inverses.
1 Introduction
This paper is concerned with the analysis of the generalized inverses of a ønitedimensional
perturbed linear operator. If the analysis of the eigenvalues and eigenvectors of a perturbed
linear operator has been carried out in detail (see e.g. [5, 22]), few results are available for
generalized inverses.
We consider the reduced resolvent of the perturbed operator, which is also known as Drazin
generalized inverse in linear algebra [7, 10]. However, as we use spectral theory and the theory
