Let
denote a
describing matrix
for
, and let
be given. We have
-
![{\displaystyle {}\chi _{M}\,(\lambda )=\det {\left(\lambda E_{n}-M\right)}=0\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81f2506e94eb4ce83959ffe0d3908a2b17fff928)
if and only if the linear mapping
-
is not
bijective
(and not
injective)
(due to
fact
and
fact).
This is, because of
fact
and
fact,
equivalent with
-
![{\displaystyle {}\operatorname {Eig} _{\lambda }{\left(\varphi \right)}=\operatorname {ker} {\left((\lambda \operatorname {Id} _{V}-\varphi )\right)}\neq 0\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/afaf91ef09b6dd0fefd1223bd162b7a640148b5e)
and this means that the
eigenspace
for
is not the nullspace, thus
is an eigenvalue for
.