Linear mapping/Trigonalizable/Characterizations/1/Fact

Characterization of trigonalizable mappings

Let ${}K$ denote a field, and let ${}V$ denote a finite-dimensional vector space. Let

$\varphi \colon V\longrightarrow V$

denote a linear mapping. Then the following statements are equivalent.

1. ${}\varphi$ is trigonalizable.
2. There exists a ${}\varphi$-invariant flag.
3. The characteristic polynomial ${}\chi _{\varphi }$ splits into linear factors.
4. The minimal polynomial ${}\mu _{\varphi }$ splits into linear factors.

Proof, Write another proof