Vector space/Linear mapping/Generalized eigenspace/Definition

Generalized eigenspace

For a linear mapping ${}\varphi$ on a ${}K$-vector space ${}V$ and an eigenvalue ${}\lambda \in K$,

${}\operatorname {GeEig} _{\lambda }(\varphi )=\bigcup _{n\in \mathbb {N} }\operatorname {kern} {\left(\varphi -\lambda \operatorname {Id} \right)}^{n}\,$

is called generalized eigenspace of ${}\varphi$ for this eigenvalue.