Endomorphism/Geometric and algebraic multiplicity/Fact/Proof

Proof

Let $m={\dim }_K\left({\mathrm{Eig}}_{\lambda }\left(\phi \right)\right)$ and let $v_1,\dots ,v_m$ be a basis of this eigenspace. We complement this basis with $w_1,\dots ,w_{n-m}$ to get a basis of $V$, using fact. With respect to this basis, the describing matrix has the form

$\left(\begin{array}{cc}\lambda E_m& B\\ 0& C\end{array}\right).$

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

To fact