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 e-prints
- Pub Date:
- July 2003
- arXiv:
- arXiv:math/0307056
- Bibcode:
- 2003math......7056N
- Keywords:
-
- Mathematics - Functional Analysis;
- 15A18;
- 15A42;
- 15A48
- E-Print:
- Email address for author is nussbaum@math.rutgers.edu (submission handled by colleague)