The Geometric Origin of the Cayley-Hamilton Theorem: A Constructive Proof via Dimensional Syzygy

TL;DR

This paper presents a geometric and tensorial perspective on the Cayley-Hamilton theorem, showing it arises from fundamental dimensional constraints and providing a dimension-independent proof.

Contribution

It introduces a novel geometric framework linking the Cayley-Hamilton theorem to tensor syzygies and offers a universal proof applicable across dimensions.

Findings

The theorem originates from a tensor syzygy involving Levi-Civita symbols.

Explicit verification provided for 2D space.

Dimension-independent proof using Laplace expansion and Newton-Girard identities.

Abstract

We demonstrate that the Cayley-Hamilton theorem is a derived consequence of a more fundamental dimensional constraint: the syzygy formed by the tensor product of two Levi-Civita symbols, which vanishes identically in m-dimensional space. By shifting perspective from the tensor A to the isotropic operators that induce A's invariants through contraction, we reveal that the Cayley-Hamilton identity emerges when this vanishing operator acts on the m-fold tensor product of A. The intrinsic tensorial form of the theorem--invariant coefficients multiplying tensor powers--is inherited from the contraction structure rather than imposed ad hoc. We provide explicit verification for two-dimensional space and a dimension-independent proof using Laplace expansion combined with Newton-Girard identities. This framework clarifies why the theorem's structure depends on ambient dimension and suggests extensions to higher-order tensors where classical characteristic polynomial methods fail.