 
Summary: Perturbation of null spaces with application to the eigenvalue
problem and generalized inverses
Konstantin E. Avrachenkov \Lambda and Moshe Haviv y
September 27, 2000
Abstract
We consider properties of a null space of an analytically perturbed matrix. In particular,
we obtain Taylor expansions for the eigenvectors which constitute a basis for the perturbed
null space. Furthermore, we apply these results to the calculation of Puiseux expansion of the
perturbed eigenvectors in the case of general eigenvalue problem as well as to the calculation
of Laurent series expansions for the perturbed group inverse and pseudoinverse matrices.
1 Introduction
The primary goal of this paper is to analyse the null space of an analytically perturbed matrix
A('') = A 0 + ''A 1 + '' 2 A 2 + \Delta \Delta \Delta ; (1)
with A k 2 R n\Thetan ; k = 0; 1; :::, when the above series converges in a region 0 Ÿ j''j Ÿ '' max for
some positive '' max . Then, we present two applications of our results. Firstly, the results on the
perturbation of null spaces can be immediately applied to calculate the Puiseux series expansion
of the general perturbed eigenvalue problem [5, 7, 8, 17, 19, 20, 21, 22, 23]
A('')x('') = –('')x(''):
Secondly, we show how our analysis coupled with techniques used for the inversion of singular
perturbed matrices [3] can be used in order to compute the Laurent series expansion for the
