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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 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 paper'5 from 1993 by Sergey Erin 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 031:, this miniweb has the following conditions for its scores: 51:; = avg/1+ 2:4/2. 3:2 = 0:1/3, :03 = 921/3 + 202/2 + 34/2, 9:4 = 31/3 + 32/2, or. equivalently. the eigenvalue equation Ln: = :1: (Le. the eigenvalue is 1), where a: E R4. and 0 0 11/2 1/3 0 0 0 1/3 1/2 0 1/2 ' (5) 1/3 1/2 0 0 Thus. the solution of the eigenvalue problem L9: = :0 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 L 6 RM\" is a matrix with all non-negative entries. such that each column sum of L is equal to 1, Le, 2;;113-13 = l for all k = 1... . ,n. In the following problems, we study properties of these matrices. L: (a) [Stochastic matrices. 3+3pt| Let L 6 RM" be a column-stochastic matrix. i. Show that the column vector e 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