Nilpotent endomorphism/Minimal polynomial/Fact

Let ${}K$ denote a field, and let ${}V$ denote a ${}K$-vector space of finite dimension. Let

$\varphi \colon V\longrightarrow V$

be a linear mapping. Then the following statements are equivalent.

1. ${}\varphi$ is nilpotent
2. The minimal polynomial of ${}\varphi$ is a power of ${}X$.
3. The characteristic polynomial of ${}\varphi$ is a power of ${}X$.

Proof, Write another proof