One relatively new application of eigenvalues, eigenvectors, and a number of results and techniques in matrix computations and numerical analysis is on the problem of ranking web pages or documents. When we enter a query, say on Google's famous search engine, a long list of pages or web documents are returned as relevant, that is, documents that match or are related to the query in some way (in Section 3.5, we study how relevant documents are matched with a given query). However, the order in which these pages appear on our screen is far from arbitrary; they follow a determined order according to a certain degree of importance. For instance, if we enter the query “news”, and the first web pages on the list are cnn, msnbc, newsgoogle, foxnews, newsyahoo, and abcnews, that means that those news sources besides being related to the query, they are supposed to be more important than hundreds or thousands of other web pages related to news. But, how is this rank of importance determined? In particular, how is this done, taking into account the enormous size of the World Wide Web?

As we will see in this chapter, the mathematical modeling of the problem of ranking web pages leads to the problem of computing an eigenvector associated with the eigenvalue of largest magnitude. Obviously, this requires not only an efficient numerical method for this purpose, but also, given the special case of the huge matrix resulting from the modeling, it requires above all a ...

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