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

Xiao Wang

TL;DR

The paper explains the dimension-dependent form of the Cayley-Hamilton theorem by deriving it from the fundamental syzygy $\boldsymbol{\varepsilon} \otimes \boldsymbol{\varepsilon} = 0$ in $m$-dimensional space. It reframes the problem by treating isotropic contraction operators as the parameters and the tensor $A$ as the variable, showing that CH emerges when the vanishing operator acts on $A^{\otimes m}$; a dimension-independent Laplace expansion with Newton–Girard identities provides a general proof, complemented by explicit two-dimensional verification. This approach clarifies why CH coefficients are invariants multiplying tensor powers and highlights the ambient dimension as a structural constraint rather than an ad hoc insertion. The methodology extends to higher-order tensors where classical characteristic-polynomial methods fail and strengthens connections to invariant theory and representation-theoretic frameworks in rational mechanics.

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.

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

TL;DR

The paper explains the dimension-dependent form of the Cayley-Hamilton theorem by deriving it from the fundamental syzygy in -dimensional space. It reframes the problem by treating isotropic contraction operators as the parameters and the tensor as the variable, showing that CH emerges when the vanishing operator acts on ; a dimension-independent Laplace expansion with Newton–Girard identities provides a general proof, complemented by explicit two-dimensional verification. This approach clarifies why CH coefficients are invariants multiplying tensor powers and highlights the ambient dimension as a structural constraint rather than an ad hoc insertion. The methodology extends to higher-order tensors where classical characteristic-polynomial methods fail and strengthens connections to invariant theory and representation-theoretic frameworks in rational mechanics.

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.
Paper Structure (15 sections, 1 theorem, 22 equations)

This paper contains 15 sections, 1 theorem, 22 equations.

Table of Contents

  1. Introduction
  2. Mathematical Preliminaries
  3. The Cayley-Hamilton Theorem
  4. Isotropic Tensors and Contraction Control
  5. The Levi-Civita Symbol and Fundamental Syzygy
  6. The Cayley-Hamilton Theorem as a Derived Syzygy
  7. From Variable to Parameter: The Quotient Rule
  8. The Fundamental Syzygy Induces Cayley-Hamilton
  9. Explicit Verification: Two-Dimensional Space
  10. Discussion and Outlook
  11. Dimension-Independent Derivation of Cayley-Hamilton from $\delta^{(m+1)} = 0$
  12. Principal Invariants via Generalized Kronecker Delta
  13. Laplace Expansion of the $(m+1)$-th Order Delta
  14. Connection to Newton-Girard Identities
  15. Completion of the Proof

Key Result

Proposition 1

The Cayley-Hamilton theorem is the explicit form of the identity $(\varepsilon \otimes \varepsilon) : A^{\otimes m} = 0$ when indices are assigned to yield a second-order tensor output. Its intrinsic form---invariant coefficients multiplying tensor powers---is inherited from the tensorial structure

Theorems & Definitions (3)

  • Remark 1
  • Proposition 1
  • Remark 2