A linear transformation on a finite dimentional nontrivial vector space has at least one eigenvalue.

Theorem: Every linear transformation (T: V o V) on a finite-dimensional nontrivial vector space ((\dim V \geq 1)) over an algebraically closed field (such as (\mathbb{C})) has at least one eigenvalue.

Proof sketch: The characteristic polynomial (p_T(\lambda) = \det(A - \lambda I)) is a polynomial of degree (n = \dim V \geq 1). By the Fundamental Theorem of Algebra, every non-constant polynomial over (\mathbb{C}) has at least one root. That root is an eigenvalue.

Important caveat: Over (\mathbb{R}), this may fail. For example, the rotation (T(x,y) = (-y, x)) in (\mathbb{R}^2) has characteristic polynomial (\lambda^2 + 1), which has no real roots. The same transformation over (\mathbb{C}) has eigenvalues (i) and (-i).

Consequence: This theorem guarantees that eigenvalue analysis is always possible when working over (\mathbb{C}).