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5. The next problems di5cuss eigenvalues and eigenvectors ofstochastic matriCes. Stochastic matrices are very useful for ranking sport teams or the importance of web pages. In factI Google's famous PageRank algorithm5 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 paper6 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 timesI and Google is worth more than 500.000,000,000 USD todayT. 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 21,, this mini-web has the following conditions for its scores: 331= 133/1 +334/2, $2 = 31/3,$:1= x1/3+x2/2+x4/2, 1'4 = 271/3+1'2/21 or, equivalently, the eigenvalue equation Ln: = a: (i.e.. the eigenvalue is 1), where a: E R4, and 0 0 11/2 1/3 0 0 0 L: 1/3 1/2 0 1/2 ' (5) 1/3 1/2 0 0 Thus, the solution of the eigenvalue problem La: :1: 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-stoohastic matrix L E RM" is a matrix with all non-negative entries. such that each column sum of L is equal to 1I i.e., 2;?=1ljk = l for all k = 1,. . . ,n. In the following problems. we study properties of these matrices. (a) [Stochastic matrices. 3+3pt] Let L 6 RM" be a column-stochastic matrix. i. Show that the column vector (-3 of all ones is an eigenvector of LT. 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 11 corresponding to the eigenvalue 1 of L either have all positive or all negative values.3 We prove this by contradiction. i. Suppose an eigenvector 'v E R\" corresponding to the eigenvalue 1 has negative and positive components. Show that for every 1' we have N I'Uz'l