Endomorphism/Invariant linear subspace/Minimal polynomial/Divisibility/Fact

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

φ : V ⟶ V

be a linear mapping. Let $U \subseteq V$ be a $φ$ -invariant linear subspace and

φ |_{U} : U ⟶ U

the restriction to $U$ (also in the target).

Then the

of

φ

is a multiple of the minimal polynomial of

φ |_{U}

.