{
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  "candidateOnly": false,
  "category": "Linear algebra",
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  "collectionSlug": "landmark-theorems-in-mathlib",
  "exactFormalStatement": "theorem LinearMap.aeval_self_charpoly {R : Type*} {M : Type*} [CommRing R] [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M] (f : M →ₗ[R] M) : (Polynomial.aeval f) f.charpoly = 0",
  "family": "Characteristic polynomials",
  "id": "library.mathlib.cayley-hamilton-theorem.v001",
  "links": {
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    "collectionData": "/data/collections/mathlib-landmarks.json",
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    "entryData": "/data/library-theorems/cayley-hamilton-theorem.json",
    "evidence": "/proofs/artifact.library.mathlib.cayley-hamilton-theorem.v001.html",
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  "landmark": {
    "completeFormalType": "theorem LinearMap.aeval_self_charpoly {R : Type*} {M : Type*} [CommRing R] [AddCommGroup M] [Module R M] [Module.Free R M] [Module.Finite R M] (f : M →ₗ[R] M) : (Polynomial.aeval f) f.charpoly = 0",
    "concepts": [
      "endomorphism",
      "characteristic polynomial",
      "polynomial evaluation",
      "finite free module"
    ],
    "plainLanguage": "Take a linear transformation of a finite free module, form its characteristic polynomial, and substitute the transformation itself into that polynomial. The resulting endomorphism is zero.",
    "poster": {
      "alt": "An ivory linear-algebra poster shows an unlabeled finite module lattice, cobalt endomorphism arcs, blank gold polynomial-combination layers, the formula chi sub f of f equals zero, and exact module scope.",
      "byteSize": 2511124,
      "caption": "A finite-free-module endomorphism satisfies its own characteristic polynomial.",
      "description": "The poster distinguishes an endomorphism from the zero polynomial evaluation, shows the matrix-mediated transformation route without field-only language, and states the commutative-ring finite-free-module scope.",
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      "generatedAt": "2026-07-22T01:26:27Z",
      "height": 1536,
      "mediaType": "image/png",
      "modelRequested": "gpt-image-2",
      "publicPath": "/assets/library-theorems/cayley-hamilton-theorem/theorem-poster-v1.png",
      "sha256": "sha256:4ad9f96b75150a624e7765195d06d0923a25fa6f887982c84e569ae722378679",
      "title": "Cayley–Hamilton theorem at a glance",
      "width": 1024,
      "kind": "theorem_poster",
      "transcript": "LINEAR ALGEBRA · CHARACTERISTIC POLYNOMIALS\nCAYLEY–HAMILTON THEOREM\nLet f be an endomorphism\nof a finite free module\nχ_f(f) = 0\nAN ENDOMORPHISM SATISFIES ITS OWN CHARACTERISTIC POLYNOMIAL\nEXACT SCOPE\nCommutative base ring and finite free module. Not restricted to matrices over a field."
    },
    "visual": {
      "alt": "A repeated sixteen-node module lattice passes through cobalt endomorphism arcs and gold polynomial layers before collapsing to sixteen calm empty positions.",
      "byteSize": 2478733,
      "caption": "The characteristic polynomial combines powers of the endomorphism into the zero map.",
      "description": "A fixed finite lattice is preserved across repeated transformation layers, then the characteristic-polynomial combination resolves to an empty lattice of zero outputs without implying that the original endomorphism is zero.",
      "derivatives": [
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          "height": 427,
          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/cayley-hamilton-theorem/theorem-schematic-v1-640.webp",
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          "publicPath": "/assets/library-theorems/cayley-hamilton-theorem/theorem-schematic-v1-1200.webp",
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      "generatedAt": "2026-07-21T23:27:34Z",
      "height": 1024,
      "mediaType": "image/png",
      "modelRequested": "gpt-image-2",
      "publicPath": "/assets/library-theorems/cayley-hamilton-theorem/theorem-schematic-v1.png",
      "sha256": "sha256:a634990a2cd2d6f9228e44f6467ef4f3e25b56470f4e9f143be56334ffa1ec19",
      "title": "Cayley–Hamilton theorem schematic",
      "width": 1536
    },
    "walkthrough": null,
    "whyLandmark": "Cayley–Hamilton binds the polynomial invariants of a linear transformation back to the transformation itself. It is a cornerstone of matrix theory and module-theoretic linear algebra."
  },
  "nonClaims": [
    "Proof Atlas did not originate the Cayley–Hamilton theorem or Mathlib's declaration.",
    "The conclusion says that the characteristic polynomial evaluated at f is zero; it does not say that f itself is zero.",
    "The generated explanation and visuals are not proof evidence."
  ],
  "publicExportEligible": true,
  "relationship": "upstream_direct",
  "schemaVersion": "library-theorem-entry-data.v1",
  "statementId": "statement.library.mathlib.cayley-hamilton-theorem.v001",
  "statusAxes": {
    "acceptedAtlasResult": "Not recorded for the preferred artifact",
    "locallyReproduced": "Exact upstream declaration replayed",
    "reviewedPage": "2 of 3 presentation reviews recorded",
    "upstreamIndexed": "Pinned source bytes verified locally"
  },
  "summary": "An endomorphism of a finite free module over a commutative ring annihilates its own characteristic polynomial.",
  "targetId": "target.library.mathlib.cayley-hamilton-theorem.v001",
  "title": "Cayley–Hamilton Theorem",
  "upstreamOrigin": {
    "declarationName": "LinearMap.aeval_self_charpoly",
    "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/LinearAlgebra/Charpoly/Basic.lean#L90",
    "packageName": "mathlib",
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    "sourceFile": "Mathlib/LinearAlgebra/Charpoly/Basic.lean",
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    "verificationKind": "git_worktree"
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  "publicPresentationReview": {
    "status": "reviewed"
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}