Endomorphism/Polynomial/Eigenvector/Fact

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

${\displaystyle f\colon V\longrightarrow V}$

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

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