Endomorphism/Generalized eigenspace/Algebraic multiplicity/Fact

Let

$\varphi \colon V\longrightarrow V$

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

Then the

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

of ${}\lambda$.

Proof, Write another proof