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Theorem List for Metamath Proof Explorer - 20601-20700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmptcoe1matfsupp 20601* The mapping extracting the entries of the coefficient matrices of a polynomial over matrices at a fixed position is finitely supported. (Contributed by AV, 6-Oct-2019.) (Proof shortened by AV, 23-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝐼𝑁𝐽𝑁) → (𝑘 ∈ ℕ0 ↦ (𝐼((coe1𝑂)‘𝑘)𝐽)) finSupp (0g𝑅))
 
Theoremmply1topmatcllem 20602* Lemma for mply1topmatcl 20604. (Contributed by AV, 6-Oct-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)    &   𝑃 = (Poly1𝑅)    &    · = ( ·𝑠𝑃)    &   𝐸 = (.g‘(mulGrp‘𝑃))    &   𝑌 = (var1𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝐼𝑁𝐽𝑁) → (𝑘 ∈ ℕ0 ↦ ((𝐼((coe1𝑂)‘𝑘)𝐽) · (𝑘𝐸𝑌))) finSupp (0g𝑃))
 
Theoremmply1topmatval 20603* A polynomial over matrices transformed into a polynomial matrix. 𝐼 is the inverse function of the transformation 𝑇 of polynomial matrices into polynomials over matrices: (𝑇‘(𝐼𝑂)) = 𝑂) (see mp2pm2mp 20610). (Contributed by AV, 6-Oct-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)    &   𝑃 = (Poly1𝑅)    &    · = ( ·𝑠𝑃)    &   𝐸 = (.g‘(mulGrp‘𝑃))    &   𝑌 = (var1𝑅)    &   𝐼 = (𝑝𝐿 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))       ((𝑁𝑉𝑂𝐿) → (𝐼𝑂) = (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))
 
Theoremmply1topmatcl 20604* A polynomial over matrices transformed into a polynomial matrix is a polynomial matrix. (Contributed by AV, 6-Oct-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)    &   𝑃 = (Poly1𝑅)    &    · = ( ·𝑠𝑃)    &   𝐸 = (.g‘(mulGrp‘𝑃))    &   𝑌 = (var1𝑅)    &   𝐼 = (𝑝𝐿 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝐼𝑂) ∈ 𝐵)
 
Theoremmp2pm2mplem1 20605* Lemma 1 for mp2pm2mp 20610. (Contributed by AV, 9-Oct-2019.) (Revised by AV, 5-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)    &    · = ( ·𝑠𝑃)    &   𝐸 = (.g‘(mulGrp‘𝑃))    &   𝑌 = (var1𝑅)    &   𝐼 = (𝑝𝐿 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝐼𝑂) = (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))
 
Theoremmp2pm2mplem2 20606* Lemma 2 for mp2pm2mp 20610. (Contributed by AV, 10-Oct-2019.) (Revised by AV, 5-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)    &    · = ( ·𝑠𝑃)    &   𝐸 = (.g‘(mulGrp‘𝑃))    &   𝑌 = (var1𝑅)    &   𝐼 = (𝑝𝐿 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))    &   𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))))) ∈ 𝐵)
 
Theoremmp2pm2mplem3 20607* Lemma 3 for mp2pm2mp 20610. (Contributed by AV, 10-Oct-2019.) (Revised by AV, 5-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)    &    · = ( ·𝑠𝑃)    &   𝐸 = (.g‘(mulGrp‘𝑃))    &   𝑌 = (var1𝑅)    &   𝐼 = (𝑝𝐿 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))    &   𝑃 = (Poly1𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝐾 ∈ ℕ0) → ((𝐼𝑂) decompPMat 𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌)))))‘𝐾)))
 
Theoremmp2pm2mplem4 20608* Lemma 4 for mp2pm2mp 20610. (Contributed by AV, 12-Oct-2019.) (Revised by AV, 5-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)    &    · = ( ·𝑠𝑃)    &   𝐸 = (.g‘(mulGrp‘𝑃))    &   𝑌 = (var1𝑅)    &   𝐼 = (𝑝𝐿 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))    &   𝑃 = (Poly1𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝐾 ∈ ℕ0) → ((𝐼𝑂) decompPMat 𝐾) = ((coe1𝑂)‘𝐾))
 
Theoremmp2pm2mplem5 20609* Lemma 5 for mp2pm2mp 20610. (Contributed by AV, 12-Oct-2019.) (Revised by AV, 5-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)    &    · = ( ·𝑠𝑃)    &   𝐸 = (.g‘(mulGrp‘𝑃))    &   𝑌 = (var1𝑅)    &   𝐼 = (𝑝𝐿 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))    &   𝑃 = (Poly1𝑅)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑋 = (var1𝐴)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝑘 ∈ ℕ0 ↦ (((𝐼𝑂) decompPMat 𝑘) (𝑘 𝑋))) finSupp (0g𝑄))
 
Theoremmp2pm2mp 20610* A polynomial over matrices transformed into a polynomial matrix transformed back into the polynomial over matrices. (Contributed by AV, 12-Oct-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)    &    · = ( ·𝑠𝑃)    &   𝐸 = (.g‘(mulGrp‘𝑃))    &   𝑌 = (var1𝑅)    &   𝐼 = (𝑝𝐿 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))    &   𝑃 = (Poly1𝑅)    &   𝑇 = (𝑁 pMatToMatPoly 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝑇‘(𝐼𝑂)) = 𝑂)
 
Theorempm2mpghmlem2 20611* Lemma 2 for pm2mpghm 20615. (Contributed by AV, 15-Oct-2019.) (Revised by AV, 4-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑋 = (var1𝐴)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) (𝑘 𝑋))) finSupp (0g𝑄))
 
Theorempm2mpghmlem1 20612 Lemma 1 for pm2mpghm . (Contributed by AV, 15-Oct-2019.) (Revised by AV, 4-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑋 = (var1𝐴)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝐾 ∈ ℕ0) → ((𝑀 decompPMat 𝐾) (𝐾 𝑋)) ∈ 𝐿)
 
Theorempm2mpfo 20613 The transformation of polynomial matrices into polynomials over matrices is a function mapping polynomial matrices onto polynomials over matrices. (Contributed by AV, 12-Oct-2019.) (Revised by AV, 6-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑋 = (var1𝐴)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)    &   𝑇 = (𝑁 pMatToMatPoly 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵onto𝐿)
 
Theorempm2mpf1o 20614 The transformation of polynomial matrices into polynomials over matrices is a 1-1 function mapping polynomial matrices onto polynomials over matrices. (Contributed by AV, 14-Oct-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑋 = (var1𝐴)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)    &   𝑇 = (𝑁 pMatToMatPoly 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵1-1-onto𝐿)
 
Theorempm2mpghm 20615 The transformation of polynomial matrices into polynomials over matrices is an additive group homomorphism. (Contributed by AV, 16-Oct-2019.) (Revised by AV, 6-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑋 = (var1𝐴)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)    &   𝑇 = (𝑁 pMatToMatPoly 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐶 GrpHom 𝑄))
 
Theorempm2mpgrpiso 20616 The transformation of polynomial matrices into polynomials over matrices is an additive group isomorphism. (Contributed by AV, 17-Oct-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑋 = (var1𝐴)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)    &   𝑇 = (𝑁 pMatToMatPoly 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐶 GrpIso 𝑄))
 
Theorempm2mpmhmlem1 20617* Lemma 1 for pm2mpmhm 20619. (Contributed by AV, 21-Oct-2019.) (Revised by AV, 6-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑋 = (var1𝐴)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)    &   𝑇 = (𝑁 pMatToMatPoly 𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑙 ∈ ℕ0 ↦ ((𝐴 Σg (𝑘 ∈ (0...𝑙) ↦ ((𝑥 decompPMat 𝑘)(.r𝐴)(𝑦 decompPMat (𝑙𝑘))))) (𝑙 𝑋))) finSupp (0g𝑄))
 
Theorempm2mpmhmlem2 20618* Lemma 2 for pm2mpmhm 20619. (Contributed by AV, 22-Oct-2019.) (Revised by AV, 6-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝑇 = (𝑁 pMatToMatPoly 𝑅)    &   𝐵 = (Base‘𝐶)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥𝐵𝑦𝐵 (𝑇‘(𝑥(.r𝐶)𝑦)) = ((𝑇𝑥)(.r𝑄)(𝑇𝑦)))
 
Theorempm2mpmhm 20619 The transformation of polynomial matrices into polynomials over matrices is a homomorphism of multiplicative monoids. (Contributed by AV, 22-Oct-2019.) (Revised by AV, 6-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝑇 = (𝑁 pMatToMatPoly 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ ((mulGrp‘𝐶) MndHom (mulGrp‘𝑄)))
 
Theorempm2mprhm 20620 The transformation of polynomial matrices into polynomials over matrices is a ring homomorphism. (Contributed by AV, 22-Oct-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝑇 = (𝑁 pMatToMatPoly 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐶 RingHom 𝑄))
 
Theorempm2mprngiso 20621 The transformation of polynomial matrices into polynomials over matrices is a ring isomorphism. (Contributed by AV, 22-Oct-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝑇 = (𝑁 pMatToMatPoly 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐶 RingIso 𝑄))
 
Theorempmmpric 20622 The ring of polynomial matrices over a ring is isomorphic to the ring of polynomials over matrices of the same dimension over the same ring. (Contributed by AV, 30-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶𝑟 𝑄)
 
Theoremmonmat2matmon 20623 The transformation of a polynomial matrix having scaled monomials with the same power as entries into a scaled monomial as a polynomial over matrices. (Contributed by AV, 11-Nov-2019.) (Revised by AV, 7-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑋 = (var1𝐴)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐾 = (Base‘𝐴)    &   𝑄 = (Poly1𝐴)    &   𝐼 = (𝑁 pMatToMatPoly 𝑅)    &   𝐸 = (.g‘(mulGrp‘𝑃))    &   𝑌 = (var1𝑅)    &    · = ( ·𝑠𝐶)    &   𝑇 = (𝑁 matToPolyMat 𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0)) → (𝐼‘((𝐿𝐸𝑌) · (𝑇𝑀))) = (𝑀 (𝐿 𝑋)))
 
Theorempm2mp 20624* The transformation of a sum of matrices having scaled monomials with the same power as entries into a sum of scaled monomials as a polynomial over matrices. (Contributed by AV, 12-Nov-2019.) (Revised by AV, 7-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑋 = (var1𝐴)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐾 = (Base‘𝐴)    &   𝑄 = (Poly1𝐴)    &   𝐼 = (𝑁 pMatToMatPoly 𝑅)    &   𝐸 = (.g‘(mulGrp‘𝑃))    &   𝑌 = (var1𝑅)    &    · = ( ·𝑠𝐶)    &   𝑇 = (𝑁 matToPolyMat 𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾𝑚0) ∧ 𝑀 finSupp (0g𝐴))) → (𝐼‘(𝐶 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸𝑌) · (𝑇‘(𝑀𝑛)))))) = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ ((𝑀𝑛) (𝑛 𝑋)))))
 
11.5  The characteristic polynomial

According to Wikipedia ("Characteristic polynomial", 31-Jul-2019, https://en.wikipedia.org/wiki/Characteristic_polynomial): "In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix as coefficients.". Based on the definition of the characteristic polynomial of a square matrix (df-chpmat 20626) the eigenvalues and corresponding eigenvectors can be defined.

 
11.5.1  Definition and basic properties

The characteristic polynomial of a matrix 𝐴 is the determinat of the characteristic matrix of 𝐴: (𝑡𝐼𝐴).

 
Syntaxcchpmat 20625 Extend class notation with the characteristic polynomial.
class CharPlyMat
 
Definitiondf-chpmat 20626* Define the characteristic polynomial of a square matrix. According to Wikipedia ("Characteristic polynomial", 31-Jul-2019, https://en.wikipedia.org/wiki/Characteristic_polynomial): "The characteristic polynomial of [an n x n matrix] A, denoted by pA(t), is the polynomial defined by pA ( t ) = det ( t I - A ) where I denotes the n-by-n identity matrix.". In addition, however, the underlying ring must be commutative, see definition in [Lang], p. 561: " Let k be a commutative ring ... Let M be any n x n matrix in k ... We define the characteristic polynomial PM(t) to be the determinant det ( t In - M ) where In is the unit n x n matrix." To be more precise, the matrices A and I on the right hand side are matrices with coefficients of a polynomial ring. Therefore, the original matrix A over a given commutative ring must be transformed into corresponding matrices over the polynomial ring over the given ring. (Contributed by AV, 2-Aug-2019.)
CharPlyMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ ((𝑛 maDet (Poly1𝑟))‘(((var1𝑟)( ·𝑠 ‘(𝑛 Mat (Poly1𝑟)))(1r‘(𝑛 Mat (Poly1𝑟))))(-g‘(𝑛 Mat (Poly1𝑟)))((𝑛 matToPolyMat 𝑟)‘𝑚)))))
 
Theoremchmatcl 20627 Closure of the characteristic matrix of a matrix. (Contributed by AV, 25-Oct-2019.) (Proof shortened by AV, 29-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝑋 = (var1𝑅)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &    = (-g𝑌)    &    · = ( ·𝑠𝑌)    &    1 = (1r𝑌)    &   𝐻 = ((𝑋 · 1 ) (𝑇𝑀))       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → 𝐻 ∈ (Base‘𝑌))
 
Theoremchmatval 20628 The entries of the characteristic matrix of a matrix. (Contributed by AV, 2-Aug-2019.) (Proof shortened by AV, 10-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝑋 = (var1𝑅)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &    = (-g𝑌)    &    · = ( ·𝑠𝑌)    &    1 = (1r𝑌)    &   𝐻 = ((𝑋 · 1 ) (𝑇𝑀))    &    = (-g𝑃)    &    0 = (0g𝑃)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝐼𝑁𝐽𝑁)) → (𝐼𝐻𝐽) = if(𝐼 = 𝐽, (𝑋 (𝐼(𝑇𝑀)𝐽)), ( 0 (𝐼(𝑇𝑀)𝐽))))
 
Theoremchpmatfval 20629* Value of the characteristic polynomial function. (Contributed by AV, 2-Aug-2019.)
𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝐷 = (𝑁 maDet 𝑃)    &    = (-g𝑌)    &   𝑋 = (var1𝑅)    &    · = ( ·𝑠𝑌)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &    1 = (1r𝑌)       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝐶 = (𝑚𝐵 ↦ (𝐷‘((𝑋 · 1 ) (𝑇𝑚)))))
 
Theoremchpmatval 20630 The characteristic polynomial of a (square) matrix (expressed with a determinant). (Contributed by AV, 2-Aug-2019.)
𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝐷 = (𝑁 maDet 𝑃)    &    = (-g𝑌)    &   𝑋 = (var1𝑅)    &    · = ( ·𝑠𝑌)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &    1 = (1r𝑌)       ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → (𝐶𝑀) = (𝐷‘((𝑋 · 1 ) (𝑇𝑀))))
 
Theoremchpmatply1 20631 The characteristic polynomial of a (square) matrix over a commutative ring is a polynomial, see also the following remark in [Lang], p. 561: "[the characteristic polynomial] is an element of k[t]". (Contributed by AV, 2-Aug-2019.) (Proof shortened by AV, 29-Nov-2019.)
𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝐸 = (Base‘𝑃)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝐶𝑀) ∈ 𝐸)
 
Theoremchpmatval2 20632* The characteristic polynomial of a (square) matrix (expressed with the Leibnitz formula for the determinant). (Contributed by AV, 2-Aug-2019.)
𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &    = (-g𝑌)    &   𝑋 = (var1𝑅)    &    · = ( ·𝑠𝑌)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &    1 = (1r𝑌)    &   𝐺 = (SymGrp‘𝑁)    &   𝐻 = (Base‘𝐺)    &   𝑍 = (ℤRHom‘𝑃)    &   𝑆 = (pmSgn‘𝑁)    &   𝑈 = (mulGrp‘𝑃)    &    × = (.r𝑃)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝐶𝑀) = (𝑃 Σg (𝑝𝐻 ↦ (((𝑍𝑆)‘𝑝) × (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)((𝑋 · 1 ) (𝑇𝑀))𝑥)))))))
 
Theoremchpmat0d 20633 The characteristic polynomial of the empty matrix. (Contributed by AV, 6-Aug-2019.)
𝐶 = (∅ CharPlyMat 𝑅)       (𝑅 ∈ Ring → (𝐶‘∅) = (1r‘(Poly1𝑅)))
 
Theoremchpmat1dlem 20634 Lemma for chpmat1d 20635. (Contributed by AV, 7-Aug-2019.)
𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑋 = (var1𝑅)    &    = (-g𝑃)    &   𝑆 = (algSc‘𝑃)    &   𝐺 = (𝑁 Mat 𝑃)    &   𝑇 = (𝑁 matToPolyMat 𝑅)       ((𝑅 ∈ Ring ∧ (𝑁 = {𝐼} ∧ 𝐼𝑉) ∧ 𝑀𝐵) → (𝐼((𝑋( ·𝑠𝐺)(1r𝐺))(-g𝐺)(𝑇𝑀))𝐼) = (𝑋 (𝑆‘(𝐼𝑀𝐼))))
 
Theoremchpmat1d 20635 The characteristic polynomial of a matrix with dimension 1. (Contributed by AV, 7-Aug-2019.)
𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑋 = (var1𝑅)    &    = (-g𝑃)    &   𝑆 = (algSc‘𝑃)       ((𝑅 ∈ CRing ∧ (𝑁 = {𝐼} ∧ 𝐼𝑉) ∧ 𝑀𝐵) → (𝐶𝑀) = (𝑋 (𝑆‘(𝐼𝑀𝐼))))
 
Theoremchpdmatlem0 20636 Lemma 0 for chpdmat 20640. (Contributed by AV, 18-Aug-2019.)
𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑆 = (algSc‘𝑃)    &   𝐵 = (Base‘𝐴)    &   𝑋 = (var1𝑅)    &    0 = (0g𝑅)    &   𝐺 = (mulGrp‘𝑃)    &    = (-g𝑃)    &   𝑄 = (𝑁 Mat 𝑃)    &    1 = (1r𝑄)    &    · = ( ·𝑠𝑄)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑋 · 1 ) ∈ (Base‘𝑄))
 
Theoremchpdmatlem1 20637 Lemma 1 for chpdmat 20640. (Contributed by AV, 18-Aug-2019.)
𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑆 = (algSc‘𝑃)    &   𝐵 = (Base‘𝐴)    &   𝑋 = (var1𝑅)    &    0 = (0g𝑅)    &   𝐺 = (mulGrp‘𝑃)    &    = (-g𝑃)    &   𝑄 = (𝑁 Mat 𝑃)    &    1 = (1r𝑄)    &    · = ( ·𝑠𝑄)    &   𝑍 = (-g𝑄)    &   𝑇 = (𝑁 matToPolyMat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → ((𝑋 · 1 )𝑍(𝑇𝑀)) ∈ (Base‘𝑄))
 
Theoremchpdmatlem2 20638 Lemma 2 for chpdmat 20640. (Contributed by AV, 18-Aug-2019.)
𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑆 = (algSc‘𝑃)    &   𝐵 = (Base‘𝐴)    &   𝑋 = (var1𝑅)    &    0 = (0g𝑅)    &   𝐺 = (mulGrp‘𝑃)    &    = (-g𝑃)    &   𝑄 = (𝑁 Mat 𝑃)    &    1 = (1r𝑄)    &    · = ( ·𝑠𝑄)    &   𝑍 = (-g𝑄)    &   𝑇 = (𝑁 matToPolyMat 𝑅)       ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑖𝑁) ∧ 𝑗𝑁) ∧ 𝑖𝑗) ∧ (𝑖𝑀𝑗) = 0 ) → (𝑖((𝑋 · 1 )𝑍(𝑇𝑀))𝑗) = (0g𝑃))
 
Theoremchpdmatlem3 20639 Lemma 3 for chpdmat 20640. (Contributed by AV, 18-Aug-2019.)
𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑆 = (algSc‘𝑃)    &   𝐵 = (Base‘𝐴)    &   𝑋 = (var1𝑅)    &    0 = (0g𝑅)    &   𝐺 = (mulGrp‘𝑃)    &    = (-g𝑃)    &   𝑄 = (𝑁 Mat 𝑃)    &    1 = (1r𝑄)    &    · = ( ·𝑠𝑄)    &   𝑍 = (-g𝑄)    &   𝑇 = (𝑁 matToPolyMat 𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝐾𝑁) → (𝐾((𝑋 · 1 )𝑍(𝑇𝑀))𝐾) = (𝑋 (𝑆‘(𝐾𝑀𝐾))))
 
Theoremchpdmat 20640* The characteristic polynomial of a diagonal matrix. (Contributed by AV, 18-Aug-2019.) (Proof shortened by AV, 21-Nov-2019.)
𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑆 = (algSc‘𝑃)    &   𝐵 = (Base‘𝐴)    &   𝑋 = (var1𝑅)    &    0 = (0g𝑅)    &   𝐺 = (mulGrp‘𝑃)    &    = (-g𝑃)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑀𝑗) = 0 )) → (𝐶𝑀) = (𝐺 Σg (𝑘𝑁 ↦ (𝑋 (𝑆‘(𝑘𝑀𝑘))))))
 
Theoremchpscmat 20641* The characteristic polynomial of a (nonempty!) scalar matrix. (Contributed by AV, 21-Aug-2019.)
𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑋 = (var1𝑅)    &   𝐺 = (mulGrp‘𝑃)    &    = (.g𝐺)    &   𝐷 = {𝑚 ∈ (Base‘𝐴) ∣ ∃𝑐 ∈ (Base‘𝑅)∀𝑖𝑁𝑗𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g𝑅))}    &   𝑆 = (algSc‘𝑃)    &    = (-g𝑃)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐷𝐼𝑁 ∧ ∀𝑛𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝐶𝑀) = ((#‘𝑁) (𝑋 (𝑆𝐸))))
 
Theoremchpscmat0 20642* The characteristic polynomial of a (nonempty!) scalar matrix, expressed with its diagonal element. (Contributed by AV, 21-Aug-2019.)
𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑋 = (var1𝑅)    &   𝐺 = (mulGrp‘𝑃)    &    = (.g𝐺)    &   𝐷 = {𝑚 ∈ (Base‘𝐴) ∣ ∃𝑐 ∈ (Base‘𝑅)∀𝑖𝑁𝑗𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g𝑅))}    &   𝑆 = (algSc‘𝑃)    &    = (-g𝑃)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐷𝐼𝑁 ∧ ∀𝑛𝑁 (𝑛𝑀𝑛) = (𝐼𝑀𝐼))) → (𝐶𝑀) = ((#‘𝑁) (𝑋 (𝑆‘(𝐼𝑀𝐼)))))
 
Theoremchpscmatgsumbin 20643* The characteristic polynomial of a (nonempty!) scalar matrix, expressed as finite group sum of binomials. (Contributed by AV, 2-Sep-2019.)
𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑋 = (var1𝑅)    &   𝐺 = (mulGrp‘𝑃)    &    = (.g𝐺)    &   𝐷 = {𝑚 ∈ (Base‘𝐴) ∣ ∃𝑐 ∈ (Base‘𝑅)∀𝑖𝑁𝑗𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g𝑅))}    &   𝑆 = (algSc‘𝑃)    &    = (-g𝑃)    &   𝐹 = (.g𝑃)    &   𝐻 = (mulGrp‘𝑅)    &   𝐸 = (.g𝐻)    &   𝐼 = (invg𝑅)    &    · = ( ·𝑠𝑃)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐷𝐽𝑁 ∧ ∀𝑛𝑁 (𝑛𝑀𝑛) = (𝐽𝑀𝐽))) → (𝐶𝑀) = (𝑃 Σg (𝑙 ∈ (0...(#‘𝑁)) ↦ (((#‘𝑁)C𝑙)𝐹((((#‘𝑁) − 𝑙)𝐸(𝐼‘(𝐽𝑀𝐽))) · (𝑙 𝑋))))))
 
Theoremchpscmatgsummon 20644* The characteristic polynomial of a (nonempty!) scalar matrix, expressed as finite group sum of scaled monomials. (Contributed by AV, 2-Sep-2019.)
𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑋 = (var1𝑅)    &   𝐺 = (mulGrp‘𝑃)    &    = (.g𝐺)    &   𝐷 = {𝑚 ∈ (Base‘𝐴) ∣ ∃𝑐 ∈ (Base‘𝑅)∀𝑖𝑁𝑗𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g𝑅))}    &   𝑆 = (algSc‘𝑃)    &    = (-g𝑃)    &   𝐹 = (.g𝑃)    &   𝐻 = (mulGrp‘𝑅)    &   𝐸 = (.g𝐻)    &   𝐼 = (invg𝑅)    &    · = ( ·𝑠𝑃)    &   𝑍 = (.g𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐷𝐽𝑁 ∧ ∀𝑛𝑁 (𝑛𝑀𝑛) = (𝐽𝑀𝐽))) → (𝐶𝑀) = (𝑃 Σg (𝑙 ∈ (0...(#‘𝑁)) ↦ ((((#‘𝑁)C𝑙)𝑍(((#‘𝑁) − 𝑙)𝐸(𝐼‘(𝐽𝑀𝐽)))) · (𝑙 𝑋)))))
 
Theoremchp0mat 20645 The characteristic polynomial of the zero matrix. (Contributed by AV, 18-Aug-2019.)
𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑋 = (var1𝑅)    &   𝐺 = (mulGrp‘𝑃)    &    = (.g𝐺)    &    0 = (0g𝐴)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝐶0 ) = ((#‘𝑁) 𝑋))
 
Theoremchpidmat 20646 The characteristic polynomial of the identity matrix. (Contributed by AV, 19-Aug-2019.)
𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑋 = (var1𝑅)    &   𝐺 = (mulGrp‘𝑃)    &    = (.g𝐺)    &   𝐼 = (1r𝐴)    &   𝑆 = (algSc‘𝑃)    &    1 = (1r𝑅)    &    = (-g𝑃)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝐶𝐼) = ((#‘𝑁) (𝑋 (𝑆1 ))))
 
Theoremchmaidscmat 20647 The characteristic polynomial of a matrix multiplied with the identity matrix is a scalar matrix. (Contributed by AV, 30-Oct-2019.) (Revised by AV, 19-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐸 = (Base‘𝑃)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝐾 = (Base‘𝑌)    &    · = ( ·𝑠𝑌)    &    0 = (0g𝑃)    &    1 = (1r𝑌)    &   𝑆 = (𝑁 ScMat 𝑃)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ((𝐶𝑀) · 1 ) ∈ 𝑆)
 
11.5.2  The characteristic factor function G

In this subsection the function 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛)))))))) is discussed. This function is involved in the representation of the product of the characteristic matrix of a given matrix and its adjunct as an infinite sum, see cpmadugsum 20677. Therefore, this function is called "characteristic factor function" (in short "chfacf") in the following. It plays an important role in the proof of the Cayley-Hamilton theorem, see cayhamlem1 20665, cayhamlem3 20686 and cayhamlem4 20687.

 
Theoremfvmptnn04if 20648* The function values of a mapping from the nonnegative integers with four distinct cases. (Contributed by AV, 10-Nov-2019.)
𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵))))    &   (𝜑𝑆 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑌𝑉)    &   ((𝜑𝑁 = 0) → 𝑌 = 𝑁 / 𝑛𝐴)    &   ((𝜑 ∧ 0 < 𝑁𝑁 < 𝑆) → 𝑌 = 𝑁 / 𝑛𝐵)    &   ((𝜑𝑁 = 𝑆) → 𝑌 = 𝑁 / 𝑛𝐶)    &   ((𝜑𝑆 < 𝑁) → 𝑌 = 𝑁 / 𝑛𝐷)       (𝜑 → (𝐺𝑁) = 𝑌)
 
Theoremfvmptnn04ifa 20649* The function value of a mapping from the nonnegative integers with four distinct cases for the first case. (Contributed by AV, 10-Nov-2019.)
𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵))))    &   (𝜑𝑆 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ0)       ((𝜑𝑁 = 0 ∧ 𝑁 / 𝑛𝐴𝑉) → (𝐺𝑁) = 𝑁 / 𝑛𝐴)
 
Theoremfvmptnn04ifb 20650* The function value of a mapping from the nonnegative integers with four distinct cases for the second case. (Contributed by AV, 10-Nov-2019.)
𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵))))    &   (𝜑𝑆 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ0)       ((𝜑 ∧ (0 < 𝑁𝑁 < 𝑆) ∧ 𝑁 / 𝑛𝐵𝑉) → (𝐺𝑁) = 𝑁 / 𝑛𝐵)
 
Theoremfvmptnn04ifc 20651* The function value of a mapping from the nonnegative integers with four distinct cases for the third case. (Contributed by AV, 10-Nov-2019.)
𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵))))    &   (𝜑𝑆 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ0)       ((𝜑𝑁 = 𝑆𝑁 / 𝑛𝐶𝑉) → (𝐺𝑁) = 𝑁 / 𝑛𝐶)
 
Theoremfvmptnn04ifd 20652* The function value of a mapping from the nonnegative integers with four distinct cases for the forth case. (Contributed by AV, 10-Nov-2019.)
𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵))))    &   (𝜑𝑆 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ0)       ((𝜑𝑆 < 𝑁𝑁 / 𝑛𝐷𝑉) → (𝐺𝑁) = 𝑁 / 𝑛𝐷)
 
Theoremchfacfisf 20653* The "characteristic factor function" is a function from the nonnegative integers to polynomial matrices. (Contributed by AV, 8-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &    × = (.r𝑌)    &    = (-g𝑌)    &    0 = (0g𝑌)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝐺:ℕ0⟶(Base‘𝑌))
 
Theoremchfacfisfcpmat 20654* The "characteristic factor function" is a function from the nonnegative integers to constant polynomial matrices. (Contributed by AV, 19-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &    × = (.r𝑌)    &    = (-g𝑌)    &    0 = (0g𝑌)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))    &   𝑆 = (𝑁 ConstPolyMat 𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝐺:ℕ0𝑆)
 
Theoremchfacffsupp 20655* The "characteristic factor function" is finitely supported. (Contributed by AV, 20-Nov-2019.) (Proof shortened by AV, 23-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &    × = (.r𝑌)    &    = (-g𝑌)    &    0 = (0g𝑌)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝐺 finSupp (0g𝑌))
 
Theoremchfacfscmulcl 20656* Closure of a scaled value of the "characteristic factor function". (Contributed by AV, 9-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &    × = (.r𝑌)    &    = (-g𝑌)    &    0 = (0g𝑌)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))    &   𝑋 = (var1𝑅)    &    · = ( ·𝑠𝑌)    &    = (.g‘(mulGrp‘𝑃))       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝐾 ∈ ℕ0) → ((𝐾 𝑋) · (𝐺𝐾)) ∈ (Base‘𝑌))
 
Theoremchfacfscmul0 20657* A scaled value of the "characteristic factor function" is zero almost always. (Contributed by AV, 9-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &    × = (.r𝑌)    &    = (-g𝑌)    &    0 = (0g𝑌)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))    &   𝑋 = (var1𝑅)    &    · = ( ·𝑠𝑌)    &    = (.g‘(mulGrp‘𝑃))       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝐾 ∈ (ℤ‘(𝑠 + 2))) → ((𝐾 𝑋) · (𝐺𝐾)) = 0 )
 
Theoremchfacfscmulfsupp 20658* A mapping of scaled values of the "characteristic factor function" is finitely supported. (Contributed by AV, 8-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &    × = (.r𝑌)    &    = (-g𝑌)    &    0 = (0g𝑌)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))    &   𝑋 = (var1𝑅)    &    · = ( ·𝑠𝑌)    &    = (.g‘(mulGrp‘𝑃))       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑖 ∈ ℕ0 ↦ ((𝑖 𝑋) · (𝐺𝑖))) finSupp 0 )
 
Theoremchfacfscmulgsum 20659* Breaking up a sum of values of the "characteristic factor function" scaled by a polynomial. (Contributed by AV, 9-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &    × = (.r𝑌)    &    = (-g𝑌)    &    0 = (0g𝑌)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))    &   𝑋 = (var1𝑅)    &    · = ( ·𝑠𝑌)    &    = (.g‘(mulGrp‘𝑃))    &    + = (+g𝑌)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 𝑋) · (𝐺𝑖)))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))))) + ((((𝑠 + 1) 𝑋) · (𝑇‘(𝑏𝑠))) ((𝑇𝑀) × (𝑇‘(𝑏‘0))))))
 
Theoremchfacfpmmulcl 20660* Closure of the value of the "characteristic factor function" multiplied with a constant polynomial matrix. (Contributed by AV, 23-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &    × = (.r𝑌)    &    = (-g𝑌)    &    0 = (0g𝑌)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))    &    = (.g‘(mulGrp‘𝑌))       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝐾 ∈ ℕ0) → ((𝐾 (𝑇𝑀)) × (𝐺𝐾)) ∈ (Base‘𝑌))
 
Theoremchfacfpmmul0 20661* The value of the "characteristic factor function" multiplied with a constant polynomial matrix is zero almost always. (Contributed by AV, 23-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &    × = (.r𝑌)    &    = (-g𝑌)    &    0 = (0g𝑌)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))    &    = (.g‘(mulGrp‘𝑌))       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝐾 ∈ (ℤ‘(𝑠 + 2))) → ((𝐾 (𝑇𝑀)) × (𝐺𝐾)) = 0 )
 
Theoremchfacfpmmulfsupp 20662* A mapping of values of the "characteristic factor function" multiplied with a constant polynomial matrix is finitely supported. (Contributed by AV, 23-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &    × = (.r𝑌)    &    = (-g𝑌)    &    0 = (0g𝑌)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))    &    = (.g‘(mulGrp‘𝑌))       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑖 ∈ ℕ0 ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖))) finSupp 0 )
 
Theoremchfacfpmmulgsum 20663* Breaking up a sum of values of the "characteristic factor function" multiplied with a constant polynomial matrix. (Contributed by AV, 23-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &    × = (.r𝑌)    &    = (-g𝑌)    &    0 = (0g𝑌)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))    &    = (.g‘(mulGrp‘𝑌))    &    + = (+g𝑌)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 (𝑇𝑀)) × ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))))) + ((((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑠))) ((𝑇𝑀) × (𝑇‘(𝑏‘0))))))
 
Theoremchfacfpmmulgsum2 20664* Breaking up a sum of values of the "characteristic factor function" multiplied with a constant polynomial matrix. (Contributed by AV, 23-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &    × = (.r𝑌)    &    = (-g𝑌)    &    0 = (0g𝑌)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))    &    = (.g‘(mulGrp‘𝑌))    &    + = (+g𝑌)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ (((𝑖 (𝑇𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) (((𝑖 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑖)))))) + ((((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑠))) ((𝑇𝑀) × (𝑇‘(𝑏‘0))))))
 
Theoremcayhamlem1 20665* Lemma 1 for cayleyhamilton 20689. (Contributed by AV, 11-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &    × = (.r𝑌)    &    = (-g𝑌)    &    0 = (0g𝑌)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))    &    = (.g‘(mulGrp‘𝑌))       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))) = 0 )
 
11.5.3  The Cayley-Hamilton theorem

In this section, a direct algebraic proof for the Cayley-Hamilton theorem is provided, according to Wikipedia ("Cayley-Hamilton theorem", 09-Nov-2019, https://en.wikipedia.org/wiki/Cayley%E2%80%93Hamilton_theorem, section "A direct algebraic proof" (this approach is also used for proving Lemma 1.9 in [Hefferon] p. 427):

"This proof uses just the kind of objects needed to formulate the Cayley-Hamilton theorem: matrices with polynomials as entries. The matrix (t * In - A) whose determinant is the characteristic polynomial of A is such a matrix, and since polynomials [over a commutative ring] form a commutative ring, it has an adjugate

(1) B = adj(t * In - A) .

Then, according to the right-hand fundamental relation of the adjugate, one has

(2) ( t * In - A ) x B = det(t * In - A) x In = p(t) * In .

Since B is also a matrix with polynomials in t as entries, one can, for each i, collect the coefficients of t^i in each entry to form a matrix Bi of numbers, such that one has

(3) B = sumi = 0 to (n-1) t^i Bi .

(The way the entries of B are defined makes clear that no powers higher than t^(n-1) occur). While this looks like a polynomial with matrices as coefficients, we shall not consider such a notion; it is just a way to write a matrix with polynomial entries as a linear combination of n constant matrices, and the coefficient t^i has been written to the left of the matrix to stress this point of view.

Now, one can expand the matrix product in our equation by bilinearity

(4) p(t) * In = ( t * In - A ) x B
= ( t * In - A ) x sumi = 0 to (n-1) t^i * Bi
= sumi = 0 to (n-1) t * In x t^i Bi - sumi = 0 to (n-1) A * t^i Bi
= sumi = 0 to (n-1) t^(i+1) * Bi - sumi = 0 to (n-1) t^i * A x Bi
= t^n Bn-1 + sumi = 1 to (n-1) t^i * ( Bi-1 - A x Bi ) - A x B0 .

Writing

(5) p(t) In = t^n * In + t^(n-1) * c(n-1) x In + ... + t * c1 In + c0 In ,

one obtains an equality of two matrices with polynomial entries, written as linear combinations of constant matrices with powers of t as coefficients. Such an equality can hold only if in any matrix position the entry that is multiplied by a given power t^i is the same on both sides; it follows that the constant matrices with coefficient t^i in both expressions must be equal. Writing these equations then for i from n down to 0, one finds

(6) Bn-1 = In , Bi-1 - A x Bi = ci * In for 1 <= i <= n-1 , - A x B0 = c0 * In .

Finally, multiply the equation of the coefficients of t^i from the left by A^i, and sum up:

(7) A^n Bn-1 + sumi = 1 to (n-1) ( A^i x Bi-1 - A^(i+1) x Bi ) - A x B0 = A^n + cn-1 * A^(n-1) + ... + c1 * A + c0 * In .

The left-hand sides form a telescoping sum and cancel completely; the right-hand sides add up to p(A):

(8) 0 = p(A) .

This completes the proof."

To formalize this approach, the steps mentioned in Wikipedia must be elaborated in more detail.

The first step is to formalize the preliminaries and the objective of the theorem. In Wikipedia, the Cayley-Hamilton theorem is stated as follows: "... the Cayley-Hamilton theorem ... states that every square matrix over a commutative ring ... satisfies its own characteristic equation." Or in more detail: "If A is a given n x n matrix and In is the n x n identity matrix, then the characteristic polynomial of A is defined as p(t) = det(t * In - A), where det is the determinant operation and t is a variable for a scalar element of the base ring. Since the entries of the matrix (t * In - A) are (linear or constant) polynomials in t, the determinant is also an n-th order monic polynomial in t. The Cayley-Hamilton theorem states that if one defines an analogous matrix equation, p(A), consisting of the replacement of the scalar eigenvalues t with the matrix A, then this polynomial in the matrix A results in the zero matrix,

p(A) = 0.

The powers of A, obtained by substitution from powers of t, are defined by repeated matrix multiplication; the constant term of p(t) gives a multiple of the power A^0, which is defined as the identity matrix. The theorem allows A^n to be expressed as a linear combination of the lower matrix powers of A. When the ring is a field, the Cayley-Hamilton theorem is equivalent to the statement that the minimal polynomial of a square matrix divides its characteristic polynomial."

Actually, the definition of the characteristic polynomial of a square matrix requires some attention. According to df-chpmat 20626, the characteristic polynomial of an 𝑁 x 𝑁 matrix 𝑀 over a ring 𝑅 is defined as

((𝑁 CharPlyMat 𝑅)‘𝑀) = (𝐷‘((𝑋 · 1 ) (𝑇𝑀))))

where 𝐷 = (𝑁 maDet 𝑃) is the function mapping an 𝑁 x 𝑁 matrix over the polynomial ring over the ring 𝑅 to its determinant, 𝑋 = (var1𝑅) is the variable of the polynomials over 𝑅, 1 is the 𝑁 x 𝑁 identity matrix as matrix over the polynomial ring over the ring 𝑅 (not the 𝑁 x 𝑁 identity matrix of the matrices over the ring 𝑅!) and (𝑇𝑀) = ((𝑁 matToPolyMat 𝑅)‘𝑀) is the matrix 𝑀 over a ring 𝑅 transformed into a constant matrix over the polynomial ring over the ring 𝑅. Thus · is the multiplication of a polynomial matrix with a "scalar", i.e. a polynomial (see chpmatval 20630).

By this definition, it is ensured that ((𝑋 · 1 ) (𝑇𝑀)), the matrix whose determinat is the characteristic polynomial of the matrix 𝑀, is actually a matrix over the polynomial ring over the ring 𝑅, as stated in Wikipedia ("matrix with polynomials as entries"). This matrix is called the characteristic matrix of matrix 𝑀 (see Wikipedia "Polynomial matrix", 16-Nov-2019, https://en.wikipedia.org/wiki/Polynomial_matrix). Following the notation in Wikipedia, we denote the characteristic polynomial of the matrix 𝑀, formally defined by ((𝑁 CharPlyMat 𝑅)‘𝑀) as "p(M)" in the comments. The characteristric matrix ((𝑋 · 1 ) (𝑇𝑀)) will be denoted by "c(M)", so that "p(M) = det( c(M) )".

After the preliminaries are clarified, the objective of the Cayley-Hamilton theorem must be considered. As described in Wikipedia, the matrix 𝑀 must be "inserted" into its characteristic polynomial in an appropriate way. Since every polynomial can be represented as infinite, but finitely supported sum of monomials scaled by the corresponding coefficients (see ply1coe 19660), also the characteristic polynomial can be written in this way:

p(M) = sumi ( pi * M^i )

Here, * is the scalar multiplication in the algebra of the polynomials over the ring 𝑅, and the coefficients are elements of the ring 𝑅.

By this, we can "define" the insertion of the matrix M into its characteristic polynomial by "p(M) = sum( pi * M^i)", see also cayleyhamilton1 20691. Here, * is the scalar multiplication in the algebra of the matrices over the ring 𝑅.

To prove the Cayley-Hamilton theorem, we have to show that "p(M) = 0", where 0 is the zero matrix.

In this section, the following class variables and informal identifiers (acronyms in the form "A(B)" or "AB") are used:

class variable/ informal identifier definiens explanation
𝑁 An arbitrary finite set, used as dimension for matrices
𝑅 An arbitrary (commutative) ring: 𝑅 ∈ CRing
B(R) (Base‘𝑅) Base set of (the ring) 𝑅
𝐴 (𝑁 Mat 𝑅) Algebra of 𝑁 x 𝑁 matrices over (the ring) 𝑅
𝐵 (Base‘𝐴) Base set of the algebra of 𝑁 x 𝑁 matrices, i .e. the set of all 𝑁 x 𝑁 matrices
𝑀 An arbitrary 𝑁 x 𝑁 matrix
𝑃 (Poly1𝑅) The algebra of polynomials over (the ring) 𝑅
B(P) (Base‘𝑃) Base set of the algebra of polynomials, i .e. the set of all polynomials
𝑋, XR (var1𝑅) The variable of polynomials over (the ring) 𝑅
𝑌, XA (var1𝐴) The variable of polynomials over matrices of the algebra 𝐴
(.g‘(mulGrp‘𝑃)) The group exponentiation for polynomials over (the ring) 𝑅
^ Arbitrary group exponentiation
𝑈 (algSc‘𝑃) The injection of scalars, i.e. elements of (the ring) 𝑅 into the base set of the algebra of polynomials over 𝑅
(𝑈𝑝), S(p) ((algSc‘𝑃)‘𝑝) The element 𝑝 of (the ring) 𝑅 represented as polynomial over 𝑅
𝑌 (𝑁 Mat 𝑃) Algebra of 𝑁 x 𝑁 matrices over (the polynomial ring) 𝑃 over the ring 𝑅
B(Y) (Base‘𝑌) Base set of the algebra of polynomial 𝑁 x 𝑁 matrices, i .e. the set of all polynomial 𝑁 x 𝑁 matrices
𝑄 (Poly1𝐴) Algebra of polynomials over the ring of 𝑁 x 𝑁 matrices over the ring 𝑅
B(Q) (Base‘𝑄) Base set of the algebra of polynomials over the ring of 𝑁 x 𝑁 matrices over the ring 𝑅, i .e. the set of all polynomials having 𝑁 x 𝑁 matrices as coefficients
+, + (+g𝑌) The addition of polynomial matrices
, - (-g𝑌) The subtraction of polynomial matrices
·, *Y ( ·𝑠𝑌) The multiplication of a polynomial matrix with a scalar ( i. e. a polynomial)
*A ( ·𝑠𝐴) The multiplication of a matrix with a scalar ( i. e. an element of the underlying ring)
*Q ( ·𝑠𝑄) The multiplication of a polynomial over matrices with a scalar ( i. e. a matrix)
×, xY (.r𝑌) The multiplication of polynomial matrices
xA (.r𝐴) The multiplication of matrices
xQ (.r𝑄) The multiplication of polynomials over matrices
1, 1Y (1r𝑌) The identity matrix in the algebra of polynomial matrices over 𝑅
1A (1r𝐴) The identity matrix in the algebra of matrices over 𝑅
0, 0Y (0g𝑌) The zero matrix in the algebra of matrices consisting of polynomials
𝑇 (𝑁 matToPolyMat 𝑅) The transformation of an 𝑁 x 𝑁 matrix over 𝑅 into a polynomial 𝑁 x 𝑁 matrix over 𝑅
T1(M) (𝑇𝑀) The matrix M transformed into a polynomial 𝑁 x 𝑁 matrix over 𝑅
U(M) (𝑈𝑀) The (constant) polynomial 𝑁 x 𝑁 matrix M transformed into a matrix over the ring 𝑅. Inverse function of 𝑇: (𝑇‘(𝑈𝑀)) = 𝑀 (see m2cpminvid2 20554 )
T2(M) ((𝑁 pMatToMatPoly 𝑅)‘𝑀) The polynomial 𝑁 x 𝑁 matrix M transformed into a polynomial over the 𝑁 x 𝑁 matrices over 𝑅
𝐼, c(M) ((𝑋 · 1 ) (𝑇𝑀)) The characteristic matrix of a matrix 𝑀, i.e. the matrix whose determinant is the characteristic polynomial of the matrix 𝑀
𝐶 (𝑁 CharPlyMat 𝑅) The function mapping a matrix over (a ring) 𝑅 to its characteristic polynomial
𝐾, p(M) (𝐶𝑀) The characteristic polynomial of a matrix over (a ring) 𝑅
𝐻 (𝐾 · 1 ) The scalar matrix (diagonal matrix) with the characteristic polynomial of a matrix as diagional elements
𝐽 (𝑁 maAdju 𝑃) The function mapping a matrix consisting of polynomials to its adjugate ("matrix of cofactors")
𝑊, adj(cm(M)) (𝐽𝐼) The adjugate of the characteristic matrix of the matrix 𝑀


Using this notation, we have:
  1. "c(M) e. B(Y)", or 𝐼 ∈ (Base‘𝑌), see chmatcl 20627
  2. "p(M) e. B(P)", or 𝐾 ∈ (Base‘𝑃), see chpmatply1 20631
  3. "T(M) e. B(Y)", or (𝑇𝑀) ∈ (Base‘𝑌), see mat2pmatbas 20525
  4. 𝐽:(Base‘𝑌)⟶(Base‘𝑌), see maduf 20441
  5. "adj(cm(M)) e. B(Y)", or 𝑊 ∈ (Base‘𝑌)


Following the proof shown in Wikipedia, the following steps are performed:
  1. Write 𝑊, the adjugate of the characteristic matrix, as a finite sum of scaled monomials, see pmatcollpw3fi1 20587:
    adj(cm(M)) = sumi=0 to s ( XR ^i *Y T1(b(i)) )
    where b(i) are matrices over the ring 𝑅, so T1(b(i)) are constant polynomial matrices.
    This step corresponds to (3) in Wikipedia. In contrast to Wikipedia, we write 𝑊 as finite sum of not exactly determined number of summands, which may be greater than needed (including summands of value 0). This will be sufficient to provide a representation of (𝐼 × 𝑊) as infinite, but finitely supported sum, see step 3.
  2. Write (𝐼 × 𝑊), the product of the characteristic matrix and its adjugate as finite sum of scaled monomials, see cpmadugsumfi 20676. This representation is obtained by replacing 𝑊 by the representation resulting from step 1. and performing calculation rules available for the associative algebra of matrices over polynomials over a commutative ring:
    cm(M) *Y adj(cm(M)) = sumi=0 to s ( XR ^i *Y ( T1(b(i-1)) - T1(M) xY T1(b(i)) ) ) + XR ^(s+1) *Y ( T1(b(s)) - T1(M) xY T1(b(0))
    where b(i) are matrices over 𝑅, so T1(b(i)) are constant polynomial matrices:
    cm(M) *Y adj(cm(M))
    = cm(M) *Y sumi=0 to s ( XR ^i *Y T1(b(i)) ) [see pmatcollpw3fi1 20587 (step 1.)]
    = ( ( XA *Y 1Y ) - T1(M) ) *Y sumi=0 to s ( XR ^i *Y T1(b(i)) ) [def. of cm(M)]
    = ( XA *Y 1Y ) *Y sumi=0 to s ( XR ^i *Y T1(b(i)) ) - T1(M) *Y sumi=0 to s ( XR ^i *Y T1(b(i)) ) [see rngsubdir 18594]
    = sumi=0 to s ( XR ^i *Y ( T1(b(i-1)) - T1(M) xY T1(b(i)) ) ) + XR ^(s+1) *Y ( T1(b(s)) - T1(M) xY T1(b(0)) [see cpmadugsumlemF 20675]
    This step corresponds partially to (4) in Wikipedia.
  3. Write (𝐼 × 𝑊) as infinite, but finitely supported sum of scaled monomials, see cpmadugsum 20677:
    cm(M) * adj(cm(M)) = sumi ( XR ^i *Y G(i) )
    This representation is obtained by defining a function G for the coefficients, which we call "characteristic factor function", see chfacfisf 20653, which covers the special terms and the padding with 0. G(i) is a constant polynomial matrix (see chfacfisfcpmat 20654). This step corresponds partially to (4) in Wikipedia, with summands of value 0 added.
  4. Write 𝐻 = (𝐾 · 1 ), the scalar matrix (diagonal matrix) with the characteristic polynomial of a matrix as diagional elements, as infinite, but finitely supported sum of scaled monomials. See cpmidgsum 20667:
    p(m) *Y IY = sumi ( XR ^i *Y ( S(pi) *Y IY ) )
    The proof of cpmidgsum 20667 is making use of pmatcollpwscmat 20590, because 𝐻 = (𝐾 · 1 ) is a scalar/diagonal polynomial matrix with the characteristic polynomial "p(M)" as diagonal entries (since pi is an element of the ring 𝑅, S(pi) is a (constant) polynomial). This corresponds to (5) in Wikipedia, with summands of value 0 added.
  5. Transform the sum representation of (𝐼 × 𝑊) from step 3. into polynomials over matrices:
    T2(cm(M) * adj(cm(M))) = sumi ( U(G(i)) *Q XA ^i ) [see cpmadumatpoly 20682]
    where U(G(i)) is a matrix over the ring 𝑅.
  6. Transform the sum representation of 𝐻 from step 4. into polynomials over matrices:
    T2(p(m) *Y IY) = sumi ( pi *A IA ) *Q XA ^i ) [see cpmidpmat 20672]
  7. Equate the sum representations resulting from steps 5. and 6. by using cpmadurid 20666 to obtain the equation
    sumi ( U(G(i)) *Q XA ^i ) = sumi ( pi *A IA ) *Q XA ^i ):
    sumi ( U(G(i)) *Q XA ^i )
    = T2(cm(M) * adj(cm(M))) [see step 5.]
    = T2(p(m) *Y IY) [see cpmadurid 20666]
    = sumi ( pi *A IA ) *Q XA ^i ) [see step 6.]
    Note that this step is contained in the proof of chcoeffeq 20685, see step 9. This step corresponds to the conclusion from (4) and (5) in Wikipedia, with summands of value 0 added.
  8. Compare the sum representations of step 7. to obtain the equations U(G(i)) = pi *A IA , see chcoeffeqlem 20684. This corresponds to (6) in Wikipedia. Since the coefficients of the transformed representations and the original representations are identical, the equations of the coefficients are also valid for the original representations of steps 3. and 4.
  9. Multiply the equations of the coefficients from step 8. from the left by M^i, and sum up, see chcoeffeq 20685:
    sumi ( M^i xA U(G(i)) ) = sumi ( M^i xA ( pi *A IA) )
    This corresponds to (7) in Wikipedia.
  10. Transform the right hand side of the equation in step 9. into an appropriate form, see cayhamlem3 20686:
    sumi ( pi *A M^i )
    = sumi ( M^i xA ( pi *A IA) ) [see cayhamlem2 20683]
    = sumi ( M^i xA U(G(i)) ) [see chcoeffeq 20685]
  11. Apply the theorem for telescoping sums, see telgsumfz 18381, to the sum sumi ( T1(M)^i xY G(i) ), which results in an equation to 0:
    sumi ( T1(M)^i xY G(i) ) = 0Y, see cayhamlem1 20665:
    sumi ( T1(M)^i xY G(i) )
    = sumi=1 to s ( T1(M)^i xY T1(b(i-1)) - T1(M)^(i+1) xY T1(b(i)) )
    + ( T1(M)^(s+1) xY T1(b(s)) - T1(M) xY T1(b(0)) ) [see chfacfpmmulgsum2 20664]
    = ( T1(M) xY T1(b(0)) - T1(M)^(s+1) xY T1(b(s)) ) + ( T1 M)^(s+1) xY T1(b(s)) - T1(M) xY T1(b(0)) ) [see telgsumfz 18381]
    = 0Y [see grpnpncan0 17505] This step corresponds partially to (8) in Wikipedia.
  12. Since 𝑇 is a ring homomorphism (see mat2pmatrhm 20533), the left hand side of the equation in step 10. can be transformed into a representation appropriate to apply the result of step 11., see cayhamlem4 20687:
    sumi ( pi *A M^i )
    = sumi ( M^i xA U(G(i)) ) [see cayhamlem3 20686 (step 10.)]
    = U(T1(sumi ( M^i xA U(G(i)) ))) [see m2cpminvid 20552]
    = U(sumi T1( M^i xA U(G(i)) )) [see gsummptmhm 18334]
    = U(sumi ( T1(M^i) xY T1(U(G(i))) )) [see rhmmul 18721]
    = U(sumi ( T1(M)^i xY T1(U(G(i))) )) [see mhmmulg 17577]
    = U(sumi ( T1(M)^i xY G(i) )) [see m2cpminvid2 20554 ]
  13. Finally, combine the results of steps 11. and 12., and use the fact that 𝑇 (and therefore also its inverse 𝑈) is an injective ring homomorphism (see mat2pmatf1 20528 and mat2pmatrhm 20533) to transform the equality resulting from steps 11. and 12. into the desired equation sumi ( pi *A M^i ) = 0A , see cayleyhamilton 20689 resp. cayleyhamilton0 20688:
    sumi ( pi *A M^i )
    = U(sumi ( T1(M)^i xY G(i) )) [see cayhamlem4 20687 (step 12.)]
    = U(0Y ) [see cayhamlem1 20665 (step 11.)]
    = 0A [see m2cpminv0 20560]
The transformations in steps 5., 6., 10., 12. and 13. are not mentioned in the proof provided in Wikipedia, since it makes no distinction between a matrix over a ring 𝑅 and its representation as matrix over the polynomial ring over the ring 𝑅 in general!
 
Theoremcpmadurid 20666 The right-hand fundamental relation of the adjugate (see madurid 20444) applied to the characteristic matrix of a matrix. (Contributed by AV, 25-Oct-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝑋 = (var1𝑅)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &    = (-g𝑌)    &    · = ( ·𝑠𝑌)    &    1 = (1r𝑌)    &   𝐼 = ((𝑋 · 1 ) (𝑇𝑀))    &   𝐽 = (𝑁 maAdju 𝑃)    &    × = (.r𝑌)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝐼 × (𝐽𝐼)) = ((𝐶𝑀) · 1 ))
 
Theoremcpmidgsum 20667* Representation of the identity matrix multiplied with the characteristic polynomial of a matrix as group sum. (Contributed by AV, 7-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝑋 = (var1𝑅)    &    = (.g‘(mulGrp‘𝑃))    &    · = ( ·𝑠𝑌)    &    1 = (1r𝑌)    &   𝑈 = (algSc‘𝑃)    &   𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝐾 = (𝐶𝑀)    &   𝐻 = (𝐾 · 1 )       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐻 = (𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 𝑋) · ((𝑈‘((coe1𝐾)‘𝑛)) · 1 )))))
 
Theoremcpmidgsumm2pm 20668* Representation of the identity matrix multiplied with the characteristic polynomial of a matrix as group sum with a matrix to polynomial matrix transformation. (Contributed by AV, 13-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝑋 = (var1𝑅)    &    = (.g‘(mulGrp‘𝑃))    &    · = ( ·𝑠𝑌)    &    1 = (1r𝑌)    &   𝑈 = (algSc‘𝑃)    &   𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝐾 = (𝐶𝑀)    &   𝐻 = (𝐾 · 1 )    &   𝑂 = (1r𝐴)    &    = ( ·𝑠𝐴)    &   𝑇 = (𝑁 matToPolyMat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐻 = (𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 𝑋) · (𝑇‘(((coe1𝐾)‘𝑛) 𝑂))))))
 
Theoremcpmidpmatlem1 20669* Lemma 1 for cpmidpmat 20672. (Contributed by AV, 13-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝑋 = (var1𝑅)    &    = (.g‘(mulGrp‘𝑃))    &    · = ( ·𝑠𝑌)    &    1 = (1r𝑌)    &   𝑈 = (algSc‘𝑃)    &   𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝐾 = (𝐶𝑀)    &   𝐻 = (𝐾 · 1 )    &   𝑂 = (1r𝐴)    &    = ( ·𝑠𝐴)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐺 = (𝑘 ∈ ℕ0 ↦ (((coe1𝐾)‘𝑘) 𝑂))       (𝐿 ∈ ℕ0 → (𝐺𝐿) = (((coe1𝐾)‘𝐿) 𝑂))
 
Theoremcpmidpmatlem2 20670* Lemma 2 for cpmidpmat 20672. (Contributed by AV, 14-Nov-2019.) (Proof shortened by AV, 7-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝑋 = (var1𝑅)    &    = (.g‘(mulGrp‘𝑃))    &    · = ( ·𝑠𝑌)    &    1 = (1r𝑌)    &   𝑈 = (algSc‘𝑃)    &   𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝐾 = (𝐶𝑀)    &   𝐻 = (𝐾 · 1 )    &   𝑂 = (1r𝐴)    &    = ( ·𝑠𝐴)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐺 = (𝑘 ∈ ℕ0 ↦ (((coe1𝐾)‘𝑘) 𝑂))       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐺 ∈ (𝐵𝑚0))
 
Theoremcpmidpmatlem3 20671* Lemma 3 for cpmidpmat 20672. (Contributed by AV, 14-Nov-2019.) (Proof shortened by AV, 7-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝑋 = (var1𝑅)    &    = (.g‘(mulGrp‘𝑃))    &    · = ( ·𝑠𝑌)    &    1 = (1r𝑌)    &   𝑈 = (algSc‘𝑃)    &   𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝐾 = (𝐶𝑀)    &   𝐻 = (𝐾 · 1 )    &   𝑂 = (1r𝐴)    &    = ( ·𝑠𝐴)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐺 = (𝑘 ∈ ℕ0 ↦ (((coe1𝐾)‘𝑘) 𝑂))       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐺 finSupp (0g𝐴))
 
Theoremcpmidpmat 20672* Representation of the identity matrix multiplied with the characteristic polynomial of a matrix as polynomial over the ring of matrices. (Contributed by AV, 14-Nov-2019.) (Revised by AV, 7-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝑋 = (var1𝑅)    &    = (.g‘(mulGrp‘𝑃))    &    · = ( ·𝑠𝑌)    &    1 = (1r𝑌)    &   𝑈 = (algSc‘𝑃)    &   𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝐾 = (𝐶𝑀)    &   𝐻 = (𝐾 · 1 )    &   𝑂 = (1r𝐴)    &    = ( ·𝑠𝐴)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝑊 = (Base‘𝑌)    &   𝑄 = (Poly1𝐴)    &   𝑍 = (var1𝐴)    &    = ( ·𝑠𝑄)    &   𝐸 = (.g‘(mulGrp‘𝑄))    &   𝐼 = (𝑁 pMatToMatPoly 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝐼𝐻) = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ ((((coe1𝐾)‘𝑛) 𝑂) (𝑛𝐸𝑍)))))
 
TheoremcpmadugsumlemB 20673* Lemma B for cpmadugsum 20677. (Contributed by AV, 2-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝑋 = (var1𝑅)    &    = (.g‘(mulGrp‘𝑃))    &    · = ( ·𝑠𝑌)    &    × = (.r𝑌)    &    1 = (1r𝑌)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → ((𝑋 · 1 ) × (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 𝑋) · (𝑇‘(𝑏𝑖)))))) = (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ (((𝑖 + 1) 𝑋) · (𝑇‘(𝑏𝑖))))))
 
TheoremcpmadugsumlemC 20674* Lemma C for cpmadugsum 20677. (Contributed by AV, 2-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝑋 = (var1𝑅)    &    = (.g‘(mulGrp‘𝑃))    &    · = ( ·𝑠𝑌)    &    × = (.r𝑌)    &    1 = (1r𝑌)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → ((𝑇𝑀) × (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 𝑋) · (𝑇‘(𝑏𝑖)))))) = (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 𝑋) · ((𝑇𝑀) × (𝑇‘(𝑏𝑖)))))))
 
TheoremcpmadugsumlemF 20675* Lemma F for cpmadugsum 20677. (Contributed by AV, 7-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝑋 = (var1𝑅)    &    = (.g‘(mulGrp‘𝑃))    &    · = ( ·𝑠𝑌)    &    × = (.r𝑌)    &    1 = (1r𝑌)    &    + = (+g𝑌)    &    = (-g𝑌)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (((𝑋 · 1 ) × (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 𝑋) · (𝑇‘(𝑏𝑖)))))) ((𝑇𝑀) × (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 𝑋) · (𝑇‘(𝑏𝑖))))))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))))) + ((((𝑠 + 1) 𝑋) · (𝑇‘(𝑏𝑠))) ((𝑇𝑀) × (𝑇‘(𝑏‘0))))))
 
Theoremcpmadugsumfi 20676* The product of the characteristic matrix of a given matrix and its adjunct represented as finite sum. (Contributed by AV, 7-Nov-2019.) (Proof shortened by AV, 29-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝑋 = (var1𝑅)    &    = (.g‘(mulGrp‘𝑃))    &    · = ( ·𝑠𝑌)    &    × = (.r𝑌)    &    1 = (1r𝑌)    &    + = (+g𝑌)    &    = (-g𝑌)    &   𝐼 = ((𝑋 · 1 ) (𝑇𝑀))    &   𝐽 = (𝑁 maAdju 𝑃)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵𝑚 (0...𝑠))(𝐼 × (𝐽𝐼)) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))))) + ((((𝑠 + 1) 𝑋) · (𝑇‘(𝑏𝑠))) ((𝑇𝑀) × (𝑇‘(𝑏‘0))))))
 
Theoremcpmadugsum 20677* The product of the characteristic matrix of a given matrix and its adjunct represented as an infinite sum. (Contributed by AV, 10-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝑋 = (var1𝑅)    &    = (.g‘(mulGrp‘𝑃))    &    · = ( ·𝑠𝑌)    &    × = (.r𝑌)    &    1 = (1r𝑌)    &    + = (+g𝑌)    &    = (-g𝑌)    &   𝐼 = ((𝑋 · 1 ) (𝑇𝑀))    &   𝐽 = (𝑁 maAdju 𝑃)    &    0 = (0g𝑌)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵𝑚 (0...𝑠))(𝐼 × (𝐽𝐼)) = (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 𝑋) · (𝐺𝑖)))))
 
Theoremcpmidgsum2 20678* Representation of the identity matrix multiplied with the characteristic polynomial of a matrix as another group sum. (Contributed by AV, 10-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝑋 = (var1𝑅)    &    = (.g‘(mulGrp‘𝑃))    &    · = ( ·𝑠𝑌)    &    × = (.r𝑌)    &    1 = (1r𝑌)    &    + = (+g𝑌)    &    = (-g𝑌)    &   𝐼 = ((𝑋 · 1 ) (𝑇𝑀))    &   𝐽 = (𝑁 maAdju 𝑃)    &    0 = (0g𝑌)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))    &   𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝐾 = (𝐶𝑀)    &   𝐻 = (𝐾 · 1 )       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵𝑚 (0...𝑠))𝐻 = (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 𝑋) · (𝐺𝑖)))))
 
Theoremcpmidg2sum 20679* Equality of two sums representing the identity matrix multiplied with the characteristic polynomial of a matrix. (Contributed by AV, 11-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝑋 = (var1𝑅)    &    = (.g‘(mulGrp‘𝑃))    &    · = ( ·𝑠𝑌)    &    × = (.r𝑌)    &    1 = (1r𝑌)    &    + = (+g𝑌)    &    = (-g𝑌)    &   𝐼 = ((𝑋 · 1 ) (𝑇𝑀))    &   𝐽 = (𝑁 maAdju 𝑃)    &    0 = (0g𝑌)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))    &   𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝐾 = (𝐶𝑀)    &   𝑈 = (algSc‘𝑃)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵𝑚 (0...𝑠))(𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 𝑋) · ((𝑈‘((coe1𝐾)‘𝑖)) · 1 )))) = (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 𝑋) · (𝐺𝑖)))))
 
Theoremcpmadumatpolylem1 20680* Lemma 1 for cpmadumatpoly 20682. (Contributed by AV, 20-Nov-2019.) (Revised by AV, 15-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &    × = (.r𝑌)    &    = (-g𝑌)    &    0 = (0g𝑌)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))    &   𝑆 = (𝑁 ConstPolyMat 𝑅)    &    · = ( ·𝑠𝑌)    &    1 = (1r𝑌)    &   𝑍 = (var1𝑅)    &   𝐷 = ((𝑍 · 1 ) (𝑇𝑀))    &   𝐽 = (𝑁 maAdju 𝑃)    &   𝑊 = (Base‘𝑌)    &   𝑄 = (Poly1𝐴)    &   𝑋 = (var1𝐴)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑈 = (𝑁 cPolyMatToMat 𝑅)       ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) → (𝑈𝐺) ∈ (𝐵𝑚0))
 
Theoremcpmadumatpolylem2 20681* Lemma 2 for cpmadumatpoly 20682. (Contributed by AV, 20-Nov-2019.) (Revised by AV, 15-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &    × = (.r𝑌)    &    = (-g𝑌)    &    0 = (0g𝑌)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))    &   𝑆 = (𝑁 ConstPolyMat 𝑅)    &    · = ( ·𝑠𝑌)    &    1 = (1r𝑌)    &   𝑍 = (var1𝑅)    &   𝐷 = ((𝑍 · 1 ) (𝑇𝑀))    &   𝐽 = (𝑁 maAdju 𝑃)    &   𝑊 = (Base‘𝑌)    &   𝑄 = (Poly1𝐴)    &   𝑋 = (var1𝐴)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑈 = (𝑁 cPolyMatToMat 𝑅)       ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) → (𝑈𝐺) finSupp (0g𝐴))
 
Theoremcpmadumatpoly 20682* The product of the characteristic matrix of a given matrix and its adjunct represented as a polynomial over matrices. (Contributed by AV, 20-Nov-2019.) (Revised by AV, 7-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &    × = (.r𝑌)    &    = (-g𝑌)    &    0 = (0g𝑌)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))    &   𝑆 = (𝑁 ConstPolyMat 𝑅)    &    · = ( ·𝑠𝑌)    &    1 = (1r𝑌)    &   𝑍 = (var1𝑅)    &   𝐷 = ((𝑍 · 1 ) (𝑇𝑀))    &   𝐽 = (𝑁 maAdju 𝑃)    &   𝑊 = (Base‘𝑌)    &   𝑄 = (Poly1𝐴)    &   𝑋 = (var1𝐴)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑈 = (𝑁 cPolyMatToMat 𝑅)    &   𝐼 = (𝑁 pMatToMatPoly 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵𝑚 (0...𝑠))(𝐼‘(𝐷 × (𝐽𝐷))) = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑛)) (𝑛 𝑋)))))
 
Theoremcayhamlem2 20683 Lemma for cayhamlem3 20686. (Contributed by AV, 24-Nov-2019.)
𝐾 = (Base‘𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    1 = (1r𝐴)    &    = ( ·𝑠𝐴)    &    = (.g‘(mulGrp‘𝐴))    &    · = (.r𝐴)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐻 ∈ (𝐾𝑚0) ∧ 𝐿 ∈ ℕ0)) → ((𝐻𝐿) (𝐿 𝑀)) = ((𝐿 𝑀) · ((𝐻𝐿) 1 )))
 
Theoremchcoeffeqlem 20684* Lemma for chcoeffeq 20685. (Contributed by AV, 21-Nov-2019.) (Proof shortened by AV, 7-Dec-2019.) (Revised by AV, 15-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &    × = (.r𝑌)    &    = (-g𝑌)    &    0 = (0g𝑌)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝐾 = (𝐶𝑀)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))    &   𝑊 = (Base‘𝑌)    &    1 = (1r𝐴)    &    = ( ·𝑠𝐴)    &   𝑈 = (𝑁 cPolyMatToMat 𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((𝑈‘(𝐺𝑛))( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))) = ((Poly1𝐴) Σg (𝑛 ∈ ℕ0 ↦ ((((coe1𝐾)‘𝑛) 1 )( ·𝑠 ‘(Poly1𝐴))(𝑛(.g‘(mulGrp‘(Poly1𝐴)))(var1𝐴))))) → ∀𝑛 ∈ ℕ0 (𝑈‘(𝐺𝑛)) = (((coe1𝐾)‘𝑛) 1 )))
 
Theoremchcoeffeq 20685* The coefficients of the characteristic polynomial multiplied with the identity matrix represented by (transformed) ring elements obtained from the adjunct of the characteristic matrix. (Contributed by AV, 21-Nov-2019.) (Proof shortened by AV, 8-Dec-2019.) (Revised by AV, 15-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &    × = (.r𝑌)    &    = (-g𝑌)    &    0 = (0g𝑌)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝐾 = (𝐶𝑀)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))    &   𝑊 = (Base‘𝑌)    &    1 = (1r𝐴)    &    = ( ·𝑠𝐴)    &   𝑈 = (𝑁 cPolyMatToMat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵𝑚 (0...𝑠))∀𝑛 ∈ ℕ0 (𝑈‘(𝐺𝑛)) = (((coe1𝐾)‘𝑛) 1 ))
 
Theoremcayhamlem3 20686* Lemma for cayhamlem4 20687. (Contributed by AV, 24-Nov-2019.) (Revised by AV, 15-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &    × = (.r𝑌)    &    = (-g𝑌)    &    0 = (0g𝑌)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝐾 = (𝐶𝑀)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))    &   𝑊 = (Base‘𝑌)    &    1 = (1r𝐴)    &    = ( ·𝑠𝐴)    &   𝑈 = (𝑁 cPolyMatToMat 𝑅)    &    = (.g‘(mulGrp‘𝐴))    &    · = (.r𝐴)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵𝑚 (0...𝑠))(𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1𝐾)‘𝑛) (𝑛 𝑀)))) = (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 𝑀) · (𝑈‘(𝐺𝑛))))))
 
Theoremcayhamlem4 20687* Lemma for cayleyhamilton 20689. (Contributed by AV, 25-Nov-2019.) (Revised by AV, 15-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &    × = (.r𝑌)    &    = (-g𝑌)    &    0 = (0g𝑌)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝐾 = (𝐶𝑀)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))    &   𝑊 = (Base‘𝑌)    &    1 = (1r𝐴)    &    = ( ·𝑠𝐴)    &   𝑈 = (𝑁 cPolyMatToMat 𝑅)    &    = (.g‘(mulGrp‘𝐴))    &   𝐸 = (.g‘(mulGrp‘𝑌))       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵𝑚 (0...𝑠))(𝐴 Σg (𝑛 ∈ ℕ0 ↦ (((coe1𝐾)‘𝑛) (𝑛 𝑀)))) = (𝑈‘(𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸(𝑇𝑀)) × (𝐺𝑛))))))
 
Theoremcayleyhamilton0 20688* The Cayley-Hamilton theorem: A matrix over a commutative ring "satisfies its own characteristic equation". This version of cayleyhamilton 20689 provides definitions not used in the theorem itself, but in its proof to make it clearer, more readable and shorter compared with a proof without them (see cayleyhamiltonALT 20690)! (Contributed by AV, 25-Nov-2019.) (Revised by AV, 15-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    0 = (0g𝐴)    &    1 = (1r𝐴)    &    = ( ·𝑠𝐴)    &    = (.g‘(mulGrp‘𝐴))    &   𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝐾 = (coe1‘(𝐶𝑀))    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &    × = (.r𝑌)    &    = (-g𝑌)    &   𝑍 = (0g𝑌)    &   𝑊 = (Base‘𝑌)    &   𝐸 = (.g‘(mulGrp‘𝑌))    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, (𝑍 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 𝑍, ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))    &   𝑈 = (𝑁 cPolyMatToMat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾𝑛) (𝑛 𝑀)))) = 0 )
 
Theoremcayleyhamilton 20689* The Cayley-Hamilton theorem: A matrix over a commutative ring "satisfies its own characteristic equation", see theorem 7.8 in [Roman] p. 170 (without proof!), or theorem 3.1 in [Lang] p. 561. In other words, a matrix over a commutative ring "inserted" into its characteristic polynomial results in zero. This is Metamath 100 proof #49. (Contributed by Alexander van der Vekens, 25-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    0 = (0g𝐴)    &   𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝐾 = (coe1‘(𝐶𝑀))    &    = ( ·𝑠𝐴)    &    = (.g‘(mulGrp‘𝐴))       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾𝑛) (𝑛 𝑀)))) = 0 )
 
TheoremcayleyhamiltonALT 20690* Alternate proof of cayleyhamilton 20689, the Cayley-Hamilton theorem. This proof does not use cayleyhamilton0 20688 directly, but has the same structure as the proof of cayleyhamilton0 20688. In contrast to the proof of cayleyhamilton0 20688, only the definitions required to formulate the theorem itself are used, causing the definitions used in the lemmas being expanded, which makes the proof longer and more difficult to read. (Contributed by AV, 25-Nov-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    0 = (0g𝐴)    &   𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝐾 = (coe1‘(𝐶𝑀))    &    = ( ·𝑠𝐴)    &    = (.g‘(mulGrp‘𝐴))       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐾𝑛) (𝑛 𝑀)))) = 0 )
 
Theoremcayleyhamilton1 20691* The Cayley-Hamilton theorem: A matrix over a commutative ring "satisfies its own characteristic equation", or, in other words, a matrix over a commutative ring "inserted" into its characteristic polynomial results in zero. In this variant of cayleyhamilton 20689, the meaning of "inserted" is made more transparent: If the characteristic polynomial is a polynomial with coefficients (𝐹𝑛), then a matrix over a commutative ring "inserted" into its characteristic polynomial is the sum of these coefficients multiplied with the corresponding power of the matrix. (Contributed by AV, 25-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    0 = (0g𝐴)    &   𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝐾 = (coe1‘(𝐶𝑀))    &    = ( ·𝑠𝐴)    &    = (.g‘(mulGrp‘𝐴))    &   𝐿 = (Base‘𝑅)    &   𝑋 = (var1𝑅)    &   𝑃 = (Poly1𝑅)    &    · = ( ·𝑠𝑃)    &   𝐸 = (.g‘(mulGrp‘𝑃))    &   𝑍 = (0g𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐹 ∈ (𝐿𝑚0) ∧ 𝐹 finSupp 𝑍)) → ((𝐶𝑀) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) · (𝑛𝐸𝑋)))) → (𝐴 Σg (𝑛 ∈ ℕ0 ↦ ((𝐹𝑛) (𝑛 𝑀)))) = 0 ))
 
PART 12  BASIC TOPOLOGY
 
12.1  Topology
 
12.1.1  Topological spaces

A topology on a set is a set of subsets of that set, called open sets, which satisfy certain conditions. One condition is that the whole set be an open set. Therefore, a set is recoverable from a topology on it (as its union, see toponuni 20713), and it may sometimes be more convenient to consider topologies without reference to the underlying set. This is why we define successively the class of topologies (df-top 20693), then the function which associates with a set the set of topologies on it (df-topon 20710), and finally the class of topological spaces, as extensible structures having an underlying set and a topology on it (df-topsp 20731). Of course, a topology is the same thing as a topology on a set (see toprntopon 20723).

 
12.1.1.1  Topologies
 
Syntaxctop 20692 Syntax for the class of topologies.
class Top
 
Definitiondf-top 20693* Define the class of topologies. It is a proper class (see topnex 20794). See istopg 20694 and istop2g 20695 for the corresponding characterizations, using respectively binary intersections like in this definition and nonempty finite intersections.

The final form of the definition is due to Bourbaki (Def. 1 of [BourbakiTop1] p. I.1), while the idea of defining a topology in terms of its open sets is due to Aleksandrov. For the convoluted history of the definitions of these notions, see

Gregory H. Moore, The emergence of open sets, closed sets, and limit points in analysis and topology, Historia Mathematica 35 (2008) 220--241.

(Contributed by NM, 3-Mar-2006.) (Revised by BJ, 20-Oct-2018.)

Top = {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥 𝑦𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑦𝑧) ∈ 𝑥)}
 
Theoremistopg 20694* Express the predicate "𝐽 is a topology." See istop2g 20695 for another characterization using nonempty finite intersections instead of binary intersections.

Note: In the literature, a topology is often represented by a calligraphic letter T, which resembles the letter J. This confusion may have led to J being used by some authors (e.g., K. D. Joshi, Introduction to General Topology (1983), p. 114) and it is convenient for us since we later use 𝑇 to represent linear transformations (operators). (Contributed by Stefan Allan, 3-Mar-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

(𝐽𝐴 → (𝐽 ∈ Top ↔ (∀𝑥(𝑥𝐽 𝑥𝐽) ∧ ∀𝑥𝐽𝑦𝐽 (𝑥𝑦) ∈ 𝐽)))
 
Theoremistop2g 20695* Express the predicate "𝐽 is a topology," using nonempty finite intersections instead of binary intersections as in istopg 20694. (Contributed by NM, 19-Jul-2006.)
(𝐽𝐴 → (𝐽 ∈ Top ↔ (∀𝑥(𝑥𝐽 𝑥𝐽) ∧ ∀𝑥((𝑥𝐽𝑥 ≠ ∅ ∧ 𝑥 ∈ Fin) → 𝑥𝐽))))
 
Theoremuniopn 20696 The union of a subset of a topology (that is, the union of any family of open sets of a topology) is an open set. (Contributed by Stefan Allan, 27-Feb-2006.)
((𝐽 ∈ Top ∧ 𝐴𝐽) → 𝐴𝐽)
 
Theoremiunopn 20697* The indexed union of a subset of a topology is an open set. (Contributed by NM, 5-Oct-2006.)
((𝐽 ∈ Top ∧ ∀𝑥𝐴 𝐵𝐽) → 𝑥𝐴 𝐵𝐽)
 
Theoreminopn 20698 The intersection of two open sets of a topology is an open set. (Contributed by NM, 17-Jul-2006.)
((𝐽 ∈ Top ∧ 𝐴𝐽𝐵𝐽) → (𝐴𝐵) ∈ 𝐽)
 
Theoremfitop 20699 A topology is closed under finite intersections. (Contributed by Jeff Hankins, 7-Oct-2009.)
(𝐽 ∈ Top → (fi‘𝐽) = 𝐽)
 
Theoremfiinopn 20700 The intersection of a nonempty finite family of open sets is open. (Contributed by FL, 20-Apr-2012.)
(𝐽 ∈ Top → ((𝐴𝐽𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin) → 𝐴𝐽))
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42322
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