LINEAR ALGEBRA: Therom: If λ is an eigenvalue for At, then λ is an eigenvalue for A

May 29th, 2015
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We want to make a new matrix obtained by dividing each entry by the sum of all other entriws in the same column. Problem: What if page j doesn’t link to anything? Then you get a column of zeros

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Google's Page RankCumpute the probability that a "random surfer" will end up at a particular webpage.Ned to know the number of webpages - there are tracking sitessuppose there are 5 websites: 514Arrows represent page j (beginning) links to page i23Make a 5x5 matrixA=(Ai,j)5 I,j =1 where Ai,j 1 if page j links to page i0 otherwiseFor our example?0 0 0 0 1A = 1 0 1 0 00 1 0 0 01 0 1 0 01 0 0 0 0We want to make a new matrix obtained by dividing each entry by the sum of all other entriws in the same columnProblem: What if page j doesn't link to anything? Then you get a column of zerosSolution- change all entries in that colum to one to get a matrix A'0 0 0 1 1A' = 1 0 1 1 0Now divide the entries by column sums to get B0 1 0 1 01 0 1 1 01 0 0 1 0 0 0 0 1/5 1B = 1/3 0 1/2 1/5 0We assume that our "random sufer" has a certain tolerance for just 0 1 0 1/5 0clicking on links1/3 0 1/2 1/5 01/3 0 0 1/5 0Here, we assume it is .85 This means with probability .15 (15%) our random surfer gets bored and start at a new webpageReplace B with the matrix? C i,jEi,j = matrix with 1 in (i,j) position,zero in every other portionn = number of web pagesC = .85B + .03 =(.85/3) + .03=(.85/3) + (.09/3)=(.49/3)We get? .03 .03 .03 .2 .88.94/3 .03 .455 .2 .03C= .03 .88 .03 .2 .03.94/3 .03 .455 .2 .03.94/3 .03

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