Linear mapping/Diagonalizable/Characterizations/Fact

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 diagonalizable.
2. There exists a basis ${}{\mathfrak {v}}$ of ${}V$ such that the describing matrix ${}M_{\mathfrak {v}}^{\mathfrak {v}}(\varphi )$ is a diagonal matrix.
3. For every describing matrix ${}M=M_{\mathfrak {w}}^{\mathfrak {w}}(\varphi )$ with respect to a basis ${}{\mathfrak {w}}$, there exists an invertible matrix ${}B$ such that

   $BMB^{-1}$

   is a diagonal matrix.

Proof, Write another proof