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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 Alices browsing process is captured by a matrix A

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.
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a) Suppose Alice starts with page i; then, in the next hour she will be at exactly one of the n pages, j=1,\cdots,n with probabilities A_{i1},\cdots,A_{in} respectively. Thus, the probability that Alice will be at any one of the n pages, j=1,\cdots,n is

41 +...+ Ain

But this probability is 1 because in the next hour she will definitely be at one of the n pages; this means

41 +...+ Ain-1

and this is true for all i=1,\cdots,n. Let

v=\begin{pmatrix}1\\ \cdot\\\cdot\\ 1\end{pmatrix}

Then

(Av)_i=\sum_{j=1}^nA_{ij}v_j=\sum_{j=1}^nA_{ij}=1=v_i

for all i=1,\cdots,n. This means Av=v. Thus, \lambda=1 is an eigenvalue of A.

b) We use induction on t. If t=1 then by definition the probability that Alice ends up browsing page j (starting from page i) after t hours is A_{ij}=(A^t)_{ij} . Thus, the statement in question is true for t=1.

Suppose that the statement is true for some t >1. Now suppose that Alice starts at page i. Suppose that after t hours she is at page k. Then the probability that she will be at page j in t+1 hour is (by base case and induction hypothesis)

(A^t)_{ik}A_{kj}

Since k can be any of the n pages k=1,\cdots,n, the probability that she is at page j in t+1 hour is

\sum_{k=1}^n(A^t)_{ik}A_{kj}=(A^tA)_{ij}=(A^{t+1})_{ij}

Thus, the statement in question is true for t+1. By induction, the statement is true for all t >1​​​​​​​.

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