Endomorphism/Polynomial/Eigenvector/Fact

Let ${}V$ be a finite-dimensional vector space over a field ${}K$, and let

$f\colon V\longrightarrow V$

be a linear mapping. Let ${}v\in V$ be an eigenvector of ${}f$ with eigenvalue ${}\lambda$, and let ${}P\in K[X]$ denote a polynomial.

Then

${}(P(f))(v)=P(\lambda )v\,.$

In particular, ${}v$ is an eigenvector of ${}P(f)$ with eigenvalue ${}P(\lambda )$. The vector ${}v\neq 0$ belongs to the kernel of ${}P(f)$ if and only if ${}\lambda$ is a zero of ${}P$.

Proof, Write another proof