I have no trouble calculating eigenvalues but I have a hard time understanding how to use them. I know that you can somehow calculate a bridge's self-frequency with eigenvalues but I don't know how. What I am after is, what do eigenvectors and eigenvectors mean physically or in other ways? I know if you have linear transformation then the eigenvectors are those vectors that are just scaled and not changed in direction. I think PageRank does something with eigenvectors but I don't know how, what's the idea? Some good(simple though) text problems to actually model something real would be helpful.