Endomorphism/Restriction to generalized eigenspace/Nilpotent/Example

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. For an eigenvalue ${}\lambda \in K$, the generalized eigenspace ${}H=\operatorname {GeEig} _{\lambda }(\varphi )$ has the property that the restriction of ${}\varphi -\lambda \operatorname {Id} _{V}$ to ${}H$ is nilpotent.