Endomorphism/Eigenvalue and characteristic polynomial/Fact/Proof

Proof

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

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

if and only if the linear mapping

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

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

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

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

To fact