Characteristic polynomial/Direct sum decomposition/Fact

Let ${}V$ be a finite-dimensional ${}K$-vector space, and

$\varphi \colon V\longrightarrow V$

be a linear mapping. Let

${}V=U\oplus W\,$

denote a direct sum decomposition in ${}\varphi$-invariant linear subspaces.

Then the characteristic polynomial fulfills the relation

${}\chi _{\varphi }=\chi _{\varphi {|}_{U}}\cdot \chi _{\varphi {|}_{W}}\,.$

Proof, Write another proof