Nonetheless, linear algebra is the way it's done for real applications. It's how the literature treats the subject as well, so it's important to understand. This tool helps for that goal, so I'm glad it's there.
So I feel that if the goal is to give a conceptual understanding, linear algebra is overkill. If the goal is to show how it's done in practice then this isn't terribly useful since it doesn't even show the tip of the iceberg!
[1] http://www.aclweb.org/anthology/E/E09/E09-1005.pdf It's interesting research, but it unfortunately turns out that even the best results in word sense disambiguation aren't good enough for a lot of applications. I wish I had 2 years to do nothing but play around in this field.
I believe we know exactly what the matrix is, based on the hyperlink structure itself. It's updated and displayed on the vis here based on that, too.
btw, David Austin's article on PageRank is highly recommended : http://www.ams.org/samplings/feature-column/fcarc-pagerank