C1W4 assignment - what is deeper meaning of section 3

I was able to complete the assignment but I wanted to learn more about section 3.

In this section, we find an eigenvector X_inf for our matrix P, that has eigenvalue = to 1, so that no matter how many times we transform X_inf with P we always have X_inf.

In the context of the question, X was meant to be a starting website where one of the rows is set to 1 and everything else is set to 0. When we find X_inf we are finding a vector of starting probabilities.

I’m struggling to interpret the meaning of what we accomplished in this section. My current understanding is we found a set of starting probabilities so that no matter how many n links you click, you always have the same probabilities for your n+1 click. Why is this a useful thing to know?

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Hi @martis880!

Thanks for your post and sorry for the extreme delay in answering you.

We do not find a vector of starting probabilities. We in fact find a “limit vector” that the X_i all converge to. This means that, for n large enough, X_n will be as close as X_{\text{inf}} as you want. So you may think this as the general probability of changing pages, since we may assume that the browser has being surfing through pages before, so we may assume n large enough. This idea is a part of the PageRank algorithm developed by Google. You can use this to sort the webpages.

Imagine you have found that the most important webpage for a search is webpage A. So we may start with A and compute X_inf so we sort the next webpages to show as the most likely to be accessed from page 4. Of course this is an oversimplification of the algorithm, but I think this illustrates what X_{inf} encodes.