403/503 – Eigenvectors for operators on real odd dimensional spaces

The goal of this note is to give a proof of the following result:

Theorem 1 Let be an odd dimensional vector space over and let be linear. Then admits an eigenvector.

The proof that follows is in the spirit of Axler’s textbook, so it avoids the use of determinants. However, I feel it is easier than the argument in the book, and it has the additional advantage of not depending on the fundamental theorem of algebra. In fact, the motivation for finding this argument was to avoid the use of the fundamental theorem.

The proof we present of Theorem 1 can be seen as an elaboration of the argument in the case when that we discussed in lecture. It was found by David Milovich in Facebook.