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5. The next problems discuss eigenvalues and eigenvectors of stochastic matrices. Stochastic matrices are very useful for ranking sport teams or the importance of web pages. In fact, Google's famous PageRank algorithm is based on eigenvectors of stochastic matrices and is (part of) the reason why Google's search was/is superior to other search engines. The basic algorithm is described in a paper from 1998 by Sergey Brin and Larry Page, the founders of Google, who were at that point both students at Stanford. This original paper has been cited more than 10,000 times, and Google is worth more than 500,000,000,000 USD today?. The basic idea is to give each web page a non-negative score describing its importance. This score is derived from links pointing to that page from other web pages. Links from more important web pages are more valuable as the score of each page is distributed amongst the pages it links to Let us consider an example with 4 web pages, where page 1 links to all other pages, page 2 links to pages 3 and 4, page 3 links to page 1, and page 4 links to pages 1 and 3. Denoting the scores for the ith page by Xi, this mini-web has the following conditions for its scores: 21 = x3/1+ 24/2, x2 = 31/3, 13 = 21/3+x2/2 + 24/2, 24 = x1/3 + 22/2, or, equivalently, the eigenvalue equation Lx = x (i.e., the eigenvalue is 1), where r ER, and 1 1/2 1/3 L= 1/3 1/2 0 1/2 (5) (1/3 1/20 0 Thus, the solution of the eigenvalue problem Lx = x provides the importance score for our mini-web. The matrix L has a special structure, it is a column-stochastic matrix. In general, a column-stochastic matrix Le Rnxn is a matrix with all non-negative entries, such that each column sum of L is equal to 1, i.e., 23-11jk = 1 for all k = 1,..., n. In the following problems, we study properties of these matrices. 0 0 0 0 0 (a) [Stochastic matrices, 3+3pt) Let Le Rnxn be a column-stochastic matrix. i. Show that the column vector e of all ones is an eigenvector of L?. What's the corre- sponding eigenvalue? ii. Argue that L has an eigenvector corresponding to the eigenvalue 1. Is this sufficient to get full credit? (b) [Positive stochastic matrices, 3+3+2pt] Assume that all entries of a column-stochastic matrix L are positive. We will prove that eigenvectors v corresponding to the eigenvalue 1 of L either have all positive or all negative values. We prove this by contradiction. i. Suppose an eigenvector v R corresponding the eigenvalue 1 has negative and positive components. Show that for every i we have 72 ;|. j=1 Where in this argument is the positivity of L used? 5. The next problems discuss eigenvalues and eigenvectors of stochastic matrices. Stochastic matrices are very useful for ranking sport teams or the importance of web pages. In fact, Google's famous PageRank algorithm is based on eigenvectors of stochastic matrices and is (part of) the reason why Google's search was/is superior to other search engines. The basic algorithm is described in a paper from 1998 by Sergey Brin and Larry Page, the founders of Google, who were at that point both students at Stanford. This original paper has been cited more than 10,000 times, and Google is worth more than 500,000,000,000 USD today?. The basic idea is to give each web page a non-negative score describing its importance. This score is derived from links pointing to that page from other web pages. Links from more important web pages are more valuable as the score of each page is distributed amongst the pages it links to Let us consider an example with 4 web pages, where page 1 links to all other pages, page 2 links to pages 3 and 4, page 3 links to page 1, and page 4 links to pages 1 and 3. Denoting the scores for the ith page by Xi, this mini-web has the following conditions for its scores: 21 = x3/1+ 24/2, x2 = 31/3, 13 = 21/3+x2/2 + 24/2, 24 = x1/3 + 22/2, or, equivalently, the eigenvalue equation Lx = x (i.e., the eigenvalue is 1), where r ER, and 1 1/2 1/3 L= 1/3 1/2 0 1/2 (5) (1/3 1/20 0 Thus, the solution of the eigenvalue problem Lx = x provides the importance score for our mini-web. The matrix L has a special structure, it is a column-stochastic matrix. In general, a column-stochastic matrix Le Rnxn is a matrix with all non-negative entries, such that each column sum of L is equal to 1, i.e., 23-11jk = 1 for all k = 1,..., n. In the following problems, we study properties of these matrices. 0 0 0 0 0 (a) [Stochastic matrices, 3+3pt) Let Le Rnxn be a column-stochastic matrix. i. Show that the column vector e of all ones is an eigenvector of L?. What's the corre- sponding eigenvalue? ii. Argue that L has an eigenvector corresponding to the eigenvalue 1. Is this sufficient to get full credit? (b) [Positive stochastic matrices, 3+3+2pt] Assume that all entries of a column-stochastic matrix L are positive. We will prove that eigenvectors v corresponding to the eigenvalue 1 of L either have all positive or all negative values. We prove this by contradiction. i. Suppose an eigenvector v R corresponding the eigenvalue 1 has negative and positive components. Show that for every i we have 72 ;|. j=1 Where in this argument is the positivity of L used
