Comments on "Jordan Canonical Form of the Google Matrix"

Abstract

The Google matrix is a Web hyperlink matrix which is given by $P(\alpha)=\alpha P+(1-\alpha)E$, where $P$ is a row stochastic matrix, $E$ is a row stochastic rank-one matrix, and $0<\alpha<1$. In this paper we explore the analytic expression of the Jordan canonical form and point out that a theorem due to Serra-Capizzano (cf. Theorem 2.3 in [SIAM J. Matrix Anal. Appl., 27 (2005), pp. 305-312]) can be used for estimating the condition number of the PageRank vector as a function of $\alpha$ now viewed in the complex field. Furthermore, we give insight into a more efficient scaling matrix in order to minimize the condition number.