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

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.