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TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24
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Cayley-Hamilton by the PID-module structure theorem

Statement

For every endomorphism T of a finite-dimensional vector space, its characteristic polynomial annihilates it:

χT(T)=0.

Facts & Assumptions

Given: The minimal polynomial satisfies μT(T)=0 by The annihilator set Ann⁡(T)={p∈F[x]:p(T)=0}; once existence is proved, its unique monic generator μT is the minimal polynomial.

[L1]

On a nonzero finite-dimensional space, the largest invariant factor is the minimal polynomial (On a nonzero finite-dimensional space, the largest invariant factor is the minimal polynomial).

[L2]

The product of the invariant factors is the characteristic polynomial (The product of the invariant factors is the characteristic polynomial).

Proof

technique · direct
1.1

On a nonzero space, [L1] and [L2] show that μT, the largest factor, divides the product χT; write χT=qμT. On the zero space, χT=1 and the identity endomorphism equals the zero endomorphism because there is only one map from the zero space to itself.

L1L2
2.1

Evaluating the factorization from step 1.1 gives χT(T)=q(T)μT(T)=0. The zero endomorphism and one-dimensional spaces are included.

step 1.1givenalgebra∎

Remarks

This is the module-theoretic route to Cayley-Hamilton. The published proof Cayley-Hamilton: every finite-dimensional endomorphism satisfies its characteristic polynomial, χT(T)=0 instead uses the adjugate identity for the polynomial matrix xI−A.

Depends on

Used by

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Dependency tree · two levels

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Sources