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.

${}\varphi$ is diagonalizable.
There exists a basis ${}{\mathfrak {v}}$ of ${}V$ such that the describing matrix ${}M_{\mathfrak {v}}^{\mathfrak {v}}(\varphi )$ is a diagonal matrix.
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.