PageRank Some graduate students at Stanford got the idea that this sort of method could be used to rank a group of web pages. You type in some key words, and the software identifies a vest collection of webpages containing those words. Most of these will be garbage, as far as you are concerned. The challenge, then, is to automatically find a few good (popular) webpages. The founders of Google had the software look at each webpage w in the set of results and determine which other webpages w links to. Thus, each webpage produces a vector of zeros and ones, just as we saw above with the students' lists of friends. Next, the algorithm builds a matrix whose columns are the normalized versions of these vectors. From there, it proceeds roughly the same way we did in the popularity problem. Google uses special methods to compute popularity very quickly, and then it lists for you the results from highest to lowest popularity. This all sounds like a good plan, but you have to be sure of a few things before you can build a company out of this idea. a. Does the equation Lr = r always have a solution? (Is there anything you've seen in class so far to make you think it might not?) b. Will a solution have nonnegative entries? Negative solutions might be hard to interpret. c. Is the solution unique? If not, we will have conflicting rankings? Fortunately, the founders of Google knew a handy theorem