1. Introduction

In this paper we give a simple proof of a spectral perturbation theorem for rank-one perturbed matrices of special structure, using a well known determinant identity. This work was motivated by the recent spectral analysis of the socalled Google matrix in the computation of the PageRank for the Google Web search engine [2,3,7,8,10]. The Google matrix is a positive matrix obtained by a special rank-one perturbation of a stochastic matrix which represents the hyperlink structure of the webpages. In our more general result, the unperturbed matrix is arbitrary, but the perturbation satisfies some natural condition.