Endomorphism/Trigonalizable/Direct sum/Fact

Let

$\varphi \colon V\longrightarrow V$

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

Then ${}V$ is the

direct sum of the generalized eigenspaces, that is,

${}V=\operatorname {GeEig} _{\lambda _{1}}(\varphi )\oplus \cdots \oplus \operatorname {GeEig} _{\lambda _{m}}(\varphi )\,,$

where ${}\lambda _{1},\ldots ,\lambda _{m}$ are the different eigenvalues of ${}\varphi$, and ${}\varphi$ is the direct sum of the restrictions

$\varphi _{i}=\varphi {|}_{H_{i}}\colon H_{i}\longrightarrow H_{i}$

on the generalized eigenspaces.

Proof, Write another proof