# Alice is visiting a website with n pages. The structure of the website and Alice's browsing process is captured by... Alice is visiting a website with n pages. The structure of the website and Alice's browsing process is captured by a matrix A as follows: every hour if Alice is browsing the page i, she will goes to the page j next hour with probability Aj (assume that each entry of A is non-negative, and the entries in each row add up to 1) (a) Show that λ is an eigenvalue of A. (b) If Alice starts browsing from page i, show that after t hours, the probability that she ends up browsing page j is (A )ij.

a) Suppose Alice starts with page ; then, in the next hour she will be at exactly one of the pages, with probabilities respectively. Thus, the probability that Alice will be at any one of the pages, is But this probability is because in the next hour she will definitely be at one of the pages; this means and this is true for all . Let Then for all . This means . Thus, is an eigenvalue of .

b) We use induction on . If then by definition the probability that Alice ends up browsing page (starting from page ) after hours is . Thus, the statement in question is true for .

Suppose that the statement is true for some . Now suppose that Alice starts at page . Suppose that after hours she is at page . Then the probability that she will be at page in hour is (by base case and induction hypothesis) Since can be any of the pages , the probability that she is at page in hour is Thus, the statement in question is true for . By induction, the statement is true for all ​​​​​​​.

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