Endomorphism/Diagonalizable/Algebraic and geometric multiplicity/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 ${}\varphi$ is

diagonalizable if and only if the characteristic polynomial ${}\chi _{\varphi }$ is a product of linear factors and if for every zero ${}\lambda$ with algebraic multiplicity ${}\mu _{\lambda }$, the identity

${}\mu _{\lambda }=\dim _{K}{\left(\operatorname {Eig} _{\lambda }{\left(\varphi \right)}\right)}\,$

holds.

Proof, Write another proof