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A Convex-Analytical Proof of the Fundamental Theorem of Algebra

Clément de Seguins Pazzis

Abstract

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.

A Convex-Analytical Proof of the Fundamental Theorem of Algebra

Abstract

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.

Paper Structure

This paper contains 3 sections, 10 theorems, 4 equations.

Table of Contents

  1. Appendix: A convex-analytical proof of the weak Birkhoff theorem
  2. Basics on cones
  3. Proof of the weak Birkhoff theorem

Key Result

Theorem 1

Let $C$ be a nonzero nondegenerate cone of a finite-dimensional real vector space $E$. Let $u$ be an endomorphism of $E$ such that $u(C) \subseteq C$. Then $u$ has an eigenvector in $C$, with nonnegative associated eigenvalue.

Theorems & Definitions (20)

  • Theorem 1: Weak form of Birkhoff's theorem Birkhoff
  • Theorem 2: Weak form of the spectral theorem
  • Theorem 3: Real form of the Fundamental Theorem of Algebra
  • proof : Proof of the weak form of the spectral theorem
  • proof : Reformulated proof of the weak form of the spectral theorem
  • proof : Proof of the real form of the Fundamental Theorem of Algebra
  • Theorem 4: Hahn-Banach separation theorem for closed convex subsets
  • Proposition 5
  • proof
  • Lemma 6
  • ...and 10 more