Theorem: Every square matrix over $\mathbb{C}$ has at least one eigenvalue. This follows from the Fundamental Theorem of Algebra: the characteristic polynomial $\det(A - \lambda I)$ has degree $n$ and therefore has at least one root in $\mathbb{C}$.
Theorem: Every square matrix over $\mathbb{C}$ has at least one eigenvalue. This follows from the Fundamental Theorem of Algebra: the characteristic polynomial $\det(A - \lambda I)$ has degree $n$ and therefore has at least one root in $\mathbb{C}$.