Notes on the Second Eigenvalue of the Google Matrix
Abstract
If $A$ is an $n\times n$ matrix whose $n$ eigenvalues are ordered in terms of decreasing modules, $\lambda_1  \geq \lambda_2 \geq ... \lambda_n$, it is often of interest to estimate $\frac{\lambda_2}{\lambda_1}$. If $A$ is a row stochastic matrix (so $\lambda_1 = 1$), one can use an old formula of R. L. Dobrushin to give a useful, explicit formula for $\lambda_2$. The purpose of this note is to disseminate these known results more widely and to show how they imply, as a very special case, some recent theorems of Haveliwala and Kamvar about the second eigenvalue of the Google matrix.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 July 2003
 arXiv:
 arXiv:math/0307056
 Bibcode:
 2003math......7056N
 Keywords:

 Mathematics  Functional Analysis;
 15A18;
 15A42;
 15A48
 EPrint:
 Email address for author is nussbaum@math.rutgers.edu (submission handled by colleague)