Endomorphism/Eigenvalue and characteristic polynomial/Fact/Proof

Let ${\displaystyle {}M}$ denote a describing matrix for ${\displaystyle {}\varphi }$, and let ${\displaystyle {}\lambda \in K}$ be given. We have

${\displaystyle {}\chi _{M}\,(\lambda )=\det {\left(\lambda E_{n}-M\right)}=0\,,}$

if and only if the linear mapping

${\displaystyle \lambda \operatorname {Id} _{V}-\varphi }$

is not bijective (and not injective) (due to fact and fact). This is, because of fact and fact, equivalent to

${\displaystyle {}\operatorname {Eig} _{\lambda }{\left(\varphi \right)}=\operatorname {ker} {\left((\lambda \operatorname {Id} _{V}-\varphi )\right)}\neq 0\,,}$

and this means that the eigenspace for ${\displaystyle {}\lambda }$ is not the null space, thus ${\displaystyle {}\lambda }$ is an eigenvalue for ${\displaystyle {}\varphi }$.