A Superintroduction to Google Matrices for Undergraduates

TL;DR

This paper explains the mathematical properties of Google matrices and the PageRank vector, aiming to clarify the core concepts of Google's algorithms for undergraduate students.

Contribution

It provides an accessible introduction to Google matrices and their eigenvalues, focusing on the fundamental properties relevant to PageRank.

Findings

Eigenvalues of Google matrices have magnitude at most 1

The PageRank vector is the stochastic eigenvector for eigenvalue 1

The paper clarifies core Google algorithms for undergraduates

Abstract

In this paper we consider so-called Google matrices and show that all eigenvalues ($\lambda$) of them have a fundamental property $|\lambda|\leq 1$. The stochastic eigenvector corresponding to $\lambda=1$ called the PageRank vector plays a central role in the Google's software. We study it in detail and present some important problems. The purpose of the paper is to make {\bf the heart of Google} clearer for undergraduates.