One of the most interesting things I have encountered in linear algebra are pseudospectra and their relation to toeplitz symbol functions, as well as their associated circulant matrix eigenvalues.
Below I have included plots which illustrate this beautiful relation (click to enlarge in new tab):
Shown on the left are the Symbol functions (yellow), Toeplitz eigenvalues (blue) and circulant matrix
eigenvalues (green). On the right epsilon-pseudospectra of the same matrices are shown.
Interested? More on:
Pseudospectra
Regularization