Theorem (Operators on $\mathbb{C}$-vector spaces have an eigenvalue)

If $V$ is a Finite Dimensional $\mathbb{C}$-Vector Space and $T\in\mathscr{L}(V)$, then T has an eigenvalue or $$\lambda\mbox{ is an eigenvalue of }T\iff\exists v\in V, \ v\not=0, \mbox{ s.t. }(T-\lambda I_V)(v)=0$$