A Convex-Analytical Proof of the Fundamental Theorem of Algebra

A weak version of Birkhoff’s generalization of the Perron-Frobenius theorem states that every endomorphism of a finite-dimensional real vector that leaves invariant a non-degenerate closed convex cone has an eigenvector in that cone. Here, we show that this theorem, whose proof relies only upon basic convex analysis, yields very short proofs of both the spectral theorem for selfadjoint operators of Euclidean spaces and the Fundamental Theorem of Algebra.