TITLE: Must eigenvalues be numbers?
QUESTION [52 upvotes]: This is more a conceptual question than any other kind. As far as I know, one can define matrices over arbitrary fields, and so do linear algebra in different settings than in the typical freshman-year course. 
Now, how does the concept of eigenvalues translate when doing so? Of course, a matrix need not have any eigenvalues in a given field, that I know. But do the eigenvalues need to be numbers?
There are examples of fields such as that of the rational functions. If we have a matrix over that field, can we have rational functions as eigenvalues?

REPLY [7 votes]: I think it's worth providing a motived example where we aren't dealing with real or complex numbers as our eigenvalues
The Schrödinger Equation is a PDE over the complex numbers that is used to talk about wave-forms in quantum mechanics. The Schrödinger Equation gives a description of a wave that permits many different steady-state solutions, each one corresponding to an eigenvalue/vector pair. The eigenvectors are the wave equations and the eigenvalues are the corresponding energy functions.