Endomorphism/Generalized eigenspace/Algebraic multiplicity/Fact

Let

${\displaystyle \varphi \colon V\longrightarrow V}$

be an endomorphism on the finite-dimensional ${\displaystyle {}K}$-vector space ${\displaystyle {}V}$, and let ${\displaystyle {}\lambda \in K}$.

Then the

dimension of the generalized eigenspace ${\displaystyle {}\operatorname {GeEig} _{\lambda }(\varphi )}$ equals the algebraic multiplicity

of ${\displaystyle {}\lambda }$.