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**

Of course. The definition of an eigenvalue does not require that the field in question is that of the real or complex numbers. In fact, it doesn’t even need to be a matrix. All you need is a vector space $V$ over a field $F$, and a linear mapping $$L: V\to V.$$

Then, $\lambda\in F$ is an eigenvalue of $L$ if and only if there exists a nonzero element $v\in V$ such that $L(v)=\lambda v$.

**Attribution***Source : Link , Question Author : MonadBoy , Answer Author : 5xum*