Endomorphism/Geometric and algebraic multiplicity/Fact/Proof

Proof

Let ${}m=\dim _{K}{\left(\operatorname {Eig} _{\lambda }{\left(\varphi \right)}\right)}$ and let ${}v_{1},\ldots ,v_{m}$ be a basis of this eigenspace. We complement this basis with ${}w_{1},\ldots ,w_{n-m}$ to get a basis of ${}V$, using fact. With respect to this basis, the describing matrix has the form

${\begin{pmatrix}\lambda E_{m}&B\\0&C\end{pmatrix}}.$

Ttherefore, the characteristic polynomial equals (using exercise) ${}(X-\lambda )^{m}\cdot \chi _{C}$, so that the algebraic multiplicity is at least ${}m$.

To fact