[Paper Review] Proving the existence of eigenvalues and eigenvectors by Weierstrass's theorem

Jean Van Schaftingen|arXiv (Cornell University)|Sep 29, 2011

Advanced Topics in Algebra9 references3 citations

TL;DR

This paper presents a determinant-free proof of the existence of eigenvalues and eigenvectors for linear operators on finite-dimensional complex vector spaces, using Weierstrass's theorem on continuous functions on compact sets. The proof minimizes the Rayleigh quotient-like functional $ \frac{\|T(v) - \lambda v\|}{\|v\|} $ and shows its minimum is zero, implying the existence of a nontrivial solution $ T(\bar{v}) = \bar{\lambda}\bar{v} $, thus establishing the spectral theorem without relying on polynomials or complex analysis beyond basic continuity and norms.

ABSTRACT

I propose a proof of the existence of the existence of eigenvectors and eigenvalues in the spirit of Argand's proof of the fundamental theorem of algebra. The proof only relies on Weierstrass's theorem, the definition of the inverse of a linear operator and algebraic identities.

Motivation & Objective

• To provide a proof of the existence of eigenvalues and eigenvectors that avoids reliance on determinants, characteristic polynomials, or the fundamental theorem of algebra.
• To establish the spectral theorem for finite-dimensional complex linear operators using only elementary analysis and linear algebra tools.
• To demonstrate that the existence of eigenvectors and eigenvalues can be derived from Weierstrass's theorem on continuous functions on compact sets.
• To offer a constructive, analytic alternative to traditional algebraic or spectral-theoretic proofs in linear algebra.

Proposed method

• Define the functional $ f(v, \lambda) = \|T(v) - \lambda v\| $ on $ V \setminus \{0\} \times \mathbb{C} $, normalized by $ \|v\| = 1 $, to minimize the deviation from eigenvector behavior.
• Use Weierstrass's theorem to guarantee the existence of a minimizer $ (\bar{v}, \bar{\lambda}) $, relying on coercivity from the boundedness of linear operators on finite-dimensional spaces.
• Establish a coercivity estimate: $ \|T(v) - \lambda v\| \geq (|\lambda| - C)\|v\| $, ensuring the sublevel sets are bounded and thus relatively compact.
• Prove that if the minimum value $ \bar{c} > 0 $, then $ T - \lambda I $ is invertible for all $ \lambda $ with $ |\lambda - \bar{\lambda}| < \bar{c} $, leading to a contradiction via a geometric series argument.
• Use the identity $ \sum_{j=0}^{n-1} (S - \omega^j \sigma I)^{-1} = n S^{n-1} $ for $ S = T - \lambda I $, $ \sigma = \bar{\lambda} - \lambda $, to derive a contradiction when $ \bar{c} > 0 $.
• Apply the limit $ n \to \infty $ to the inequality $ \|T(v) - \lambda v\| \leq \bar{c}\|v\| + n \left( \frac{|\lambda - \bar{\lambda}|}{\bar{c}} \right)^n \|T(v) - \lambda v\| $, showing that $ \bar{c} = 0 $ is necessary.

Experimental results

Research questions

• RQ1 Can the existence of eigenvalues and eigenvectors be proven without using the characteristic polynomial or the fundamental theorem of algebra?
• RQ2 Is it possible to establish the spectral theorem for finite-dimensional complex linear operators using only continuous minimization and elementary operator bounds?
• RQ3 Can Weierstrass's theorem on compact sets be used to prove the existence of eigenvectors via a Rayleigh quotient-type minimization?
• RQ4 What are the implications of assuming a positive lower bound on $ \|T(v) - \lambda v\| / \|v\| $ for all $ v \neq 0 $, $ \lambda \in \mathbb{C} $, in the context of linear operators?
• RQ5 How can the invertibility of $ T - \lambda I $ be analyzed uniformly over a disk in $ \mathbb{C} $ to derive a contradiction if no eigenvalue exists?

Key findings

• The existence of a minimizer $ (\bar{v}, \bar{\lambda}) $ to the functional $ \|T(v) - \lambda v\| / \|v\| $ is guaranteed by Weierstrass's theorem due to coercivity and compactness of sublevel sets.
• If the minimum value $ \bar{c} > 0 $, then $ T - \lambda I $ is invertible for all $ \lambda $ with $ |\lambda - \bar{\lambda}| < \bar{c} $, and the resolvent is uniformly bounded.
• By constructing a sum of resolvents over roots of unity and using geometric series identities, a contradiction is derived when $ \bar{c} > 0 $, showing that $ \bar{c} $ must be zero.
• The contradiction arises in the limit as $ n \to \infty $, where the term $ \left( \frac{|\lambda - \bar{\lambda}|}{\bar{c}} \right)^n \to 0 $, forcing $ \|T(v) - \lambda v\| \leq \bar{c}\|v\| $ for some $ v $, which contradicts $ \bar{c} > 0 $ unless $ \bar{c} = 0 $.
• Therefore, the minimum value of the functional is zero, implying the existence of a non-zero vector $ \bar{v} $ and $ \bar{\lambda} \in \mathbb{C} $ such that $ T(\bar{v}) = \bar{\lambda}\bar{v} $.
• The proof is entirely analytic and avoids determinants, polynomials, and complex analysis beyond continuity and norms, offering a new, elementary route to the spectral theorem.