Endomorphism/Eigenvalue and characteristic polynomial/Fact

Let ${\displaystyle {}K}$ denote a field, and let ${\displaystyle {}V}$ denote an ${\displaystyle {}n}$-dimensional vector space. Let

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

denote a linear mapping.

Then

${\displaystyle {}\lambda \in K}$ is an eigenvalue of ${\displaystyle {}\varphi }$ if and only if ${\displaystyle {}\lambda }$ is a zero of the characteristic polynomial

${\displaystyle {}\chi _{\varphi }}$.