Vector Space Automorphisms#

Theorem: Eigentheory of Automorphisms

Let \(V\) be a vector space over a field \(F\) and let \(f: V \to V\) be an automorphism.

If \(\lambda\) is an eigenvalue of \(f\), then \(\lambda \ne 0\).

Some \(\lambda \in F\) is an eigenvalue of \(f\) if and only if \(\lambda^{-1}\) is an eigenvalue of \(f^{-1}\). In this case, \(\lambda\) and \(\lambda^{-1}\) have identical eigenspaces, identical algebraic multiplicities and identical geometric multiplicities.

Proof

TODO