 
Summary: THE $25,000,000,000
EIGENVECTOR
THE LINEAR ALGEBRA BEHIND GOOGLE
KURT BRYAN AND TANYA LEISE
Abstract. Google's success derives in large part from its PageRank algorithm, which ranks the importance
of webpages according to an eigenvector of a weighted link matrix. Analysis of the PageRank formula provides a
wonderful applied topic for a linear algebra course. Instructors may assign this article as a project to more advanced
students, or spend one or two lectures presenting the material with assigned homework from the exercises. This
material also complements the discussion of Markov chains in matrix algebra. Maple and Mathematica files supporting
this material can be found at www.rosehulman.edu/bryan.
Key words. linear algebra, PageRank, eigenvector, stochastic matrix
AMS subject classifications. 1501, 15A18, 15A51
1. Introduction. When Google went online in the late 1990's, one thing that set it apart
from other search engines was that its search result listings always seemed deliver the "good stuff"
up front. With other search engines you often had to wade through screen after screen of links
to irrelevant web pages that just happened to match the search text. Part of the magic behind
Google is its PageRank algorithm, which quantitatively rates the importance of each page on the
web, allowing Google to rank the pages and thereby present to the user the more important (and
typically most relevant and helpful) pages first.
Understanding how to calculate PageRank is essential for anyone designing a web page that they
