To me, one of the most important things about the eigenvalues of a matrix is that the product of all of them equals the determinant of that matrix. When I first heard this, I had an overwhelming sensation of euphoria. Angels sang.

I'm sure there's more to be posted in this thread.

I always imagined them like: the eigenvectors of a matrix are the vectors that don't change direction when multiplied by that matrix, and the eigenvalues are how much the eigenvectors change in size.
So they're similar to fixed points.

>>149
Yes, eigenvectors as invariant linear subspaces is a very good way of interpreting them, especially if you're studying dynamical systems (see OP's pic).