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.

If ${}\varphi$ is trigonalizable is and if it is described, with respect to a basis, by the matrix ${}M$, then there exists an invertible matrix (set ${}n=\dim _{K}{\left(V\right)}$) ${}B\in \operatorname {Mat} _{n\times n}(K)$ such that ${}BMB^{-1}$ is an upper triangular matrix.

Proof, Write another proof