Endomorphism/Trigonalizable/Canonical additive decomposition/Fact

Theorem about the canonical additive decomposition

Let

$\varphi \colon V\longrightarrow V$

denote a trigonalizable ${}K$-endomorphism on the finite-dimensional ${}K$-vector space ${}V$.

Then there exists a decomposition

${}\varphi =\varphi _{\rm {diag}}+\varphi _{\rm {nil}}\,,$

where ${}\varphi _{\rm {diag}}$ is diagonalizable and ${}\varphi _{\rm {nil}}$ is nilpotent, and where

${}\varphi _{\rm {diag}}\circ \varphi _{\rm {nil}}=\varphi _{\rm {nil}}\circ \varphi _{\rm {diag}}\,$

holds.

Proof, Write another proof