Must eigenvalues be numbers?

Question

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?

Answer 1

Eigenvalues pop up all over the place. You can find eigenvalues from an ODE e.g. y''-lambday=0 will output something like lambda=sin(nx) so to answer you question, yes the can be something other that a real number, they can be complex as well. They are not specific to matrices.