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Theorem List for Metamath Proof Explorer - 22301-22400 *Has distinct variable group(s)

Type

Label

Description

Statement

Theorem

pm2mpcoe1 22301

A coefficient of the polynomial over matrices which is the result of the transformation of a polynomial matrix is the matrix consisting of the coefficients in the polynomial entries of the polynomial matrix. (Contributed by AV, 20-Oct-2019.) (Revised by AV, 5-Dec-2019.)

𝑃 = (Poly₁‘𝑅) & 𝐶 = (𝑁 Mat 𝑃) & 𝐵 = (Base‘𝐶) & ∗ = ( ·_𝑠 ‘𝑄) & ↑ = (._g‘(mulGrp‘𝑄)) & 𝑋 = (var₁‘𝐴) & 𝐴 = (𝑁 Mat 𝑅) & 𝑄 = (Poly₁‘𝐴) & 𝑇 = (𝑁 pMatToMatPoly 𝑅) ⇒ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐵 ∧ 𝐾 ∈ ℕ₀)) → ((coe₁‘(𝑇‘𝑀))‘𝐾) = (𝑀 decompPMat 𝐾))

Theorem

idpm2idmp 22302

The transformation of the identity polynomial matrix into polynomials over matrices results in the identity of the polynomials over matrices. (Contributed by AV, 18-Oct-2019.) (Revised by AV, 5-Dec-2019.)

𝑃 = (Poly₁‘𝑅) & 𝐶 = (𝑁 Mat 𝑃) & 𝐵 = (Base‘𝐶) & ∗ = ( ·_𝑠 ‘𝑄) & ↑ = (._g‘(mulGrp‘𝑄)) & 𝑋 = (var₁‘𝐴) & 𝐴 = (𝑁 Mat 𝑅) & 𝑄 = (Poly₁‘𝐴) & 𝑇 = (𝑁 pMatToMatPoly 𝑅) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑇‘(1_r‘𝐶)) = (1_r‘𝑄))

Theorem

mptcoe1matfsupp 22303*

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 = (Poly₁‘𝐴) & 𝐿 = (Base‘𝑄) ⇒ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → (𝑘 ∈ ℕ₀ ↦ (𝐼((coe₁‘𝑂)‘𝑘)𝐽)) finSupp (0_g‘𝑅))

Theorem

mply1topmatcllem 22304*

Lemma for mply1topmatcl 22306. (Contributed by AV, 6-Oct-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝑄 = (Poly₁‘𝐴) & 𝐿 = (Base‘𝑄) & 𝑃 = (Poly₁‘𝑅) & · = ( ·_𝑠 ‘𝑃) & 𝐸 = (._g‘(mulGrp‘𝑃)) & 𝑌 = (var₁‘𝑅) ⇒ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → (𝑘 ∈ ℕ₀ ↦ ((𝐼((coe₁‘𝑂)‘𝑘)𝐽) · (𝑘𝐸𝑌))) finSupp (0_g‘𝑃))

Theorem

mply1topmatval 22305*

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‘(𝐼‘𝑂)) = 𝑂) (see mp2pm2mp 22312). (Contributed by AV, 6-Oct-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝑄 = (Poly₁‘𝐴) & 𝐿 = (Base‘𝑄) & 𝑃 = (Poly₁‘𝑅) & · = ( ·_𝑠 ‘𝑃) & 𝐸 = (._g‘(mulGrp‘𝑃)) & 𝑌 = (var₁‘𝑅) & 𝐼 = (𝑝 ∈ 𝐿 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑖((coe₁‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)))))) ⇒ ((𝑁 ∈ 𝑉 ∧ 𝑂 ∈ 𝐿) → (𝐼‘𝑂) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑖((coe₁‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))

Theorem

mply1topmatcl 22306*

A polynomial over matrices transformed into a polynomial matrix is a polynomial matrix. (Contributed by AV, 6-Oct-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝑄 = (Poly₁‘𝐴) & 𝐿 = (Base‘𝑄) & 𝑃 = (Poly₁‘𝑅) & · = ( ·_𝑠 ‘𝑃) & 𝐸 = (._g‘(mulGrp‘𝑃)) & 𝑌 = (var₁‘𝑅) & 𝐼 = (𝑝 ∈ 𝐿 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑖((coe₁‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)))))) & 𝐶 = (𝑁 Mat 𝑃) & 𝐵 = (Base‘𝐶) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝐼‘𝑂) ∈ 𝐵)

Theorem

mp2pm2mplem1 22307*

Lemma 1 for mp2pm2mp 22312. (Contributed by AV, 9-Oct-2019.) (Revised by AV, 5-Dec-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝑄 = (Poly₁‘𝐴) & 𝐿 = (Base‘𝑄) & · = ( ·_𝑠 ‘𝑃) & 𝐸 = (._g‘(mulGrp‘𝑃)) & 𝑌 = (var₁‘𝑅) & 𝐼 = (𝑝 ∈ 𝐿 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑖((coe₁‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)))))) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝐼‘𝑂) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑖((coe₁‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))

Theorem

mp2pm2mplem2 22308*

Lemma 2 for mp2pm2mp 22312. (Contributed by AV, 10-Oct-2019.) (Revised by AV, 5-Dec-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝑄 = (Poly₁‘𝐴) & 𝐿 = (Base‘𝑄) & · = ( ·_𝑠 ‘𝑃) & 𝐸 = (._g‘(mulGrp‘𝑃)) & 𝑌 = (var₁‘𝑅) & 𝐼 = (𝑝 ∈ 𝐿 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑖((coe₁‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)))))) & 𝑃 = (Poly₁‘𝑅) & 𝐶 = (𝑁 Mat 𝑃) & 𝐵 = (Base‘𝐶) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑖((coe₁‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))))) ∈ 𝐵)

Theorem

mp2pm2mplem3 22309*

Lemma 3 for mp2pm2mp 22312. (Contributed by AV, 10-Oct-2019.) (Revised by AV, 5-Dec-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝑄 = (Poly₁‘𝐴) & 𝐿 = (Base‘𝑄) & · = ( ·_𝑠 ‘𝑃) & 𝐸 = (._g‘(mulGrp‘𝑃)) & 𝑌 = (var₁‘𝑅) & 𝐼 = (𝑝 ∈ 𝐿 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑖((coe₁‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)))))) & 𝑃 = (Poly₁‘𝑅) ⇒ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐾 ∈ ℕ₀) → ((𝐼‘𝑂) decompPMat 𝐾) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe₁‘(𝑃 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑖((coe₁‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌)))))‘𝐾)))

Theorem

mp2pm2mplem4 22310*

Lemma 4 for mp2pm2mp 22312. (Contributed by AV, 12-Oct-2019.) (Revised by AV, 5-Dec-2019.)

Theorem

mp2pm2mplem5 22311*

Lemma 5 for mp2pm2mp 22312. (Contributed by AV, 12-Oct-2019.) (Revised by AV, 5-Dec-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝑄 = (Poly₁‘𝐴) & 𝐿 = (Base‘𝑄) & · = ( ·_𝑠 ‘𝑃) & 𝐸 = (._g‘(mulGrp‘𝑃)) & 𝑌 = (var₁‘𝑅) & 𝐼 = (𝑝 ∈ 𝐿 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑖((coe₁‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)))))) & 𝑃 = (Poly₁‘𝑅) & ∗ = ( ·_𝑠 ‘𝑄) & ↑ = (._g‘(mulGrp‘𝑄)) & 𝑋 = (var₁‘𝐴) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝑘 ∈ ℕ₀ ↦ (((𝐼‘𝑂) decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))) finSupp (0_g‘𝑄))

Theorem

mp2pm2mp 22312*

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 = (Poly₁‘𝐴) & 𝐿 = (Base‘𝑄) & · = ( ·_𝑠 ‘𝑃) & 𝐸 = (._g‘(mulGrp‘𝑃)) & 𝑌 = (var₁‘𝑅) & 𝐼 = (𝑝 ∈ 𝐿 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σ_g (𝑘 ∈ ℕ₀ ↦ ((𝑖((coe₁‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)))))) & 𝑃 = (Poly₁‘𝑅) & 𝑇 = (𝑁 pMatToMatPoly 𝑅) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝑇‘(𝐼‘𝑂)) = 𝑂)

Theorem

pm2mpghmlem2 22313*

Lemma 2 for pm2mpghm 22317. (Contributed by AV, 15-Oct-2019.) (Revised by AV, 4-Dec-2019.)

𝑃 = (Poly₁‘𝑅) & 𝐶 = (𝑁 Mat 𝑃) & 𝐵 = (Base‘𝐶) & ∗ = ( ·_𝑠 ‘𝑄) & ↑ = (._g‘(mulGrp‘𝑄)) & 𝑋 = (var₁‘𝐴) & 𝐴 = (𝑁 Mat 𝑅) & 𝑄 = (Poly₁‘𝐴) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑘 ∈ ℕ₀ ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))) finSupp (0_g‘𝑄))

Theorem

pm2mpghmlem1 22314

Lemma 1 for pm2mpghm . (Contributed by AV, 15-Oct-2019.) (Revised by AV, 4-Dec-2019.)

𝑃 = (Poly₁‘𝑅) & 𝐶 = (𝑁 Mat 𝑃) & 𝐵 = (Base‘𝐶) & ∗ = ( ·_𝑠 ‘𝑄) & ↑ = (._g‘(mulGrp‘𝑄)) & 𝑋 = (var₁‘𝐴) & 𝐴 = (𝑁 Mat 𝑅) & 𝑄 = (Poly₁‘𝐴) & 𝐿 = (Base‘𝑄) ⇒ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) → ((𝑀 decompPMat 𝐾) ∗ (𝐾 ↑ 𝑋)) ∈ 𝐿)

Theorem

pm2mpfo 22315

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 = (Poly₁‘𝑅) & 𝐶 = (𝑁 Mat 𝑃) & 𝐵 = (Base‘𝐶) & ∗ = ( ·_𝑠 ‘𝑄) & ↑ = (._g‘(mulGrp‘𝑄)) & 𝑋 = (var₁‘𝐴) & 𝐴 = (𝑁 Mat 𝑅) & 𝑄 = (Poly₁‘𝐴) & 𝐿 = (Base‘𝑄) & 𝑇 = (𝑁 pMatToMatPoly 𝑅) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵–onto→𝐿)

Theorem

pm2mpf1o 22316

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 = (Poly₁‘𝑅) & 𝐶 = (𝑁 Mat 𝑃) & 𝐵 = (Base‘𝐶) & ∗ = ( ·_𝑠 ‘𝑄) & ↑ = (._g‘(mulGrp‘𝑄)) & 𝑋 = (var₁‘𝐴) & 𝐴 = (𝑁 Mat 𝑅) & 𝑄 = (Poly₁‘𝐴) & 𝐿 = (Base‘𝑄) & 𝑇 = (𝑁 pMatToMatPoly 𝑅) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵–1-1-onto→𝐿)

Theorem

pm2mpghm 22317

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 = (Poly₁‘𝑅) & 𝐶 = (𝑁 Mat 𝑃) & 𝐵 = (Base‘𝐶) & ∗ = ( ·_𝑠 ‘𝑄) & ↑ = (._g‘(mulGrp‘𝑄)) & 𝑋 = (var₁‘𝐴) & 𝐴 = (𝑁 Mat 𝑅) & 𝑄 = (Poly₁‘𝐴) & 𝐿 = (Base‘𝑄) & 𝑇 = (𝑁 pMatToMatPoly 𝑅) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐶 GrpHom 𝑄))

Theorem

pm2mpgrpiso 22318

The transformation of polynomial matrices into polynomials over matrices is an additive group isomorphism. (Contributed by AV, 17-Oct-2019.)

𝑃 = (Poly₁‘𝑅) & 𝐶 = (𝑁 Mat 𝑃) & 𝐵 = (Base‘𝐶) & ∗ = ( ·_𝑠 ‘𝑄) & ↑ = (._g‘(mulGrp‘𝑄)) & 𝑋 = (var₁‘𝐴) & 𝐴 = (𝑁 Mat 𝑅) & 𝑄 = (Poly₁‘𝐴) & 𝐿 = (Base‘𝑄) & 𝑇 = (𝑁 pMatToMatPoly 𝑅) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐶 GrpIso 𝑄))

Theorem

pm2mpmhmlem1 22319*

Lemma 1 for pm2mpmhm 22321. (Contributed by AV, 21-Oct-2019.) (Revised by AV, 6-Dec-2019.)

𝑃 = (Poly₁‘𝑅) & 𝐶 = (𝑁 Mat 𝑃) & 𝐵 = (Base‘𝐶) & ∗ = ( ·_𝑠 ‘𝑄) & ↑ = (._g‘(mulGrp‘𝑄)) & 𝑋 = (var₁‘𝐴) & 𝐴 = (𝑁 Mat 𝑅) & 𝑄 = (Poly₁‘𝐴) & 𝐿 = (Base‘𝑄) & 𝑇 = (𝑁 pMatToMatPoly 𝑅) ⇒ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑙 ∈ ℕ₀ ↦ ((𝐴 Σ_g (𝑘 ∈ (0...𝑙) ↦ ((𝑥 decompPMat 𝑘)(._r‘𝐴)(𝑦 decompPMat (𝑙 − 𝑘))))) ∗ (𝑙 ↑ 𝑋))) finSupp (0_g‘𝑄))

Theorem

pm2mpmhmlem2 22320*

Lemma 2 for pm2mpmhm 22321. (Contributed by AV, 22-Oct-2019.) (Revised by AV, 6-Dec-2019.)

𝑃 = (Poly₁‘𝑅) & 𝐶 = (𝑁 Mat 𝑃) & 𝐴 = (𝑁 Mat 𝑅) & 𝑄 = (Poly₁‘𝐴) & 𝑇 = (𝑁 pMatToMatPoly 𝑅) & 𝐵 = (Base‘𝐶) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑇‘(𝑥(._r‘𝐶)𝑦)) = ((𝑇‘𝑥)(._r‘𝑄)(𝑇‘𝑦)))

Theorem

pm2mpmhm 22321

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 = (Poly₁‘𝑅) & 𝐶 = (𝑁 Mat 𝑃) & 𝐴 = (𝑁 Mat 𝑅) & 𝑄 = (Poly₁‘𝐴) & 𝑇 = (𝑁 pMatToMatPoly 𝑅) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ ((mulGrp‘𝐶) MndHom (mulGrp‘𝑄)))

Theorem

pm2mprhm 22322

The transformation of polynomial matrices into polynomials over matrices is a ring homomorphism. (Contributed by AV, 22-Oct-2019.)

𝑃 = (Poly₁‘𝑅) & 𝐶 = (𝑁 Mat 𝑃) & 𝐴 = (𝑁 Mat 𝑅) & 𝑄 = (Poly₁‘𝐴) & 𝑇 = (𝑁 pMatToMatPoly 𝑅) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐶 RingHom 𝑄))

Theorem

pm2mprngiso 22323

The transformation of polynomial matrices into polynomials over matrices is a ring isomorphism. (Contributed by AV, 22-Oct-2019.)

𝑃 = (Poly₁‘𝑅) & 𝐶 = (𝑁 Mat 𝑃) & 𝐴 = (𝑁 Mat 𝑅) & 𝑄 = (Poly₁‘𝐴) & 𝑇 = (𝑁 pMatToMatPoly 𝑅) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐶 RingIso 𝑄))

Theorem

pmmpric 22324

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 = (Poly₁‘𝑅) & 𝐶 = (𝑁 Mat 𝑃) & 𝐴 = (𝑁 Mat 𝑅) & 𝑄 = (Poly₁‘𝐴) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ≃_𝑟 𝑄)

Theorem

monmat2matmon 22325

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 = (Poly₁‘𝑅) & 𝐶 = (𝑁 Mat 𝑃) & 𝐵 = (Base‘𝐶) & ∗ = ( ·_𝑠 ‘𝑄) & ↑ = (._g‘(mulGrp‘𝑄)) & 𝑋 = (var₁‘𝐴) & 𝐴 = (𝑁 Mat 𝑅) & 𝐾 = (Base‘𝐴) & 𝑄 = (Poly₁‘𝐴) & 𝐼 = (𝑁 pMatToMatPoly 𝑅) & 𝐸 = (._g‘(mulGrp‘𝑃)) & 𝑌 = (var₁‘𝑅) & · = ( ·_𝑠 ‘𝐶) & 𝑇 = (𝑁 matToPolyMat 𝑅) ⇒ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ₀)) → (𝐼‘((𝐿𝐸𝑌) · (𝑇‘𝑀))) = (𝑀 ∗ (𝐿 ↑ 𝑋)))

Theorem

pm2mp 22326*

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 = (Poly₁‘𝑅) & 𝐶 = (𝑁 Mat 𝑃) & 𝐵 = (Base‘𝐶) & ∗ = ( ·_𝑠 ‘𝑄) & ↑ = (._g‘(mulGrp‘𝑄)) & 𝑋 = (var₁‘𝐴) & 𝐴 = (𝑁 Mat 𝑅) & 𝐾 = (Base‘𝐴) & 𝑄 = (Poly₁‘𝐴) & 𝐼 = (𝑁 pMatToMatPoly 𝑅) & 𝐸 = (._g‘(mulGrp‘𝑃)) & 𝑌 = (var₁‘𝑅) & · = ( ·_𝑠 ‘𝐶) & 𝑇 = (𝑁 matToPolyMat 𝑅) ⇒ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑_m ℕ₀) ∧ 𝑀 finSupp (0_g‘𝐴))) → (𝐼‘(𝐶 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛)))))) = (𝑄 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑀‘𝑛) ∗ (𝑛 ↑ 𝑋)))))

11.7 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 22328) the eigenvalues and corresponding eigenvectors can be defined.

11.7.1 Definition and basic properties

The characteristic polynomial of a matrix 𝐴 is the determinant of the characteristic matrix of 𝐴: (𝑡𝐼 − 𝐴).

Syntax

cchpmat 22327

Extend class notation with the characteristic polynomial.

class CharPlyMat

Definition

df-chpmat 22328*

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 p_A(t), is the polynomial defined by p_A ( 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 P_M(t) to be the determinant det ( t I_n - M ) where I_n 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 (Poly₁‘𝑟))‘(((var₁‘𝑟)( ·_𝑠 ‘(𝑛 Mat (Poly₁‘𝑟)))(1_r‘(𝑛 Mat (Poly₁‘𝑟))))(-_g‘(𝑛 Mat (Poly₁‘𝑟)))((𝑛 matToPolyMat 𝑟)‘𝑚)))))

Theorem

chmatcl 22329

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 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & 𝑋 = (var₁‘𝑅) & 𝑇 = (𝑁 matToPolyMat 𝑅) & − = (-_g‘𝑌) & · = ( ·_𝑠 ‘𝑌) & 1 = (1_r‘𝑌) & 𝐻 = ((𝑋 · 1 ) − (𝑇‘𝑀)) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝐻 ∈ (Base‘𝑌))

Theorem

chmatval 22330

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 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & 𝑋 = (var₁‘𝑅) & 𝑇 = (𝑁 matToPolyMat 𝑅) & − = (-_g‘𝑌) & · = ( ·_𝑠 ‘𝑌) & 1 = (1_r‘𝑌) & 𝐻 = ((𝑋 · 1 ) − (𝑇‘𝑀)) & ∼ = (-_g‘𝑃) & 0 = (0_g‘𝑃) ⇒ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼𝐻𝐽) = if(𝐼 = 𝐽, (𝑋 ∼ (𝐼(𝑇‘𝑀)𝐽)), ( 0 ∼ (𝐼(𝑇‘𝑀)𝐽))))

Theorem

chpmatfval 22331*

Value of the characteristic polynomial function. (Contributed by AV, 2-Aug-2019.)

𝐶 = (𝑁 CharPlyMat 𝑅) & 𝐴 = (𝑁 Mat 𝑅) & 𝐵 = (Base‘𝐴) & 𝑃 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & 𝐷 = (𝑁 maDet 𝑃) & − = (-_g‘𝑌) & 𝑋 = (var₁‘𝑅) & · = ( ·_𝑠 ‘𝑌) & 𝑇 = (𝑁 matToPolyMat 𝑅) & 1 = (1_r‘𝑌) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐶 = (𝑚 ∈ 𝐵 ↦ (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑚)))))

Theorem

chpmatval 22332

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 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & 𝐷 = (𝑁 maDet 𝑃) & − = (-_g‘𝑌) & 𝑋 = (var₁‘𝑅) & · = ( ·_𝑠 ‘𝑌) & 𝑇 = (𝑁 matToPolyMat 𝑅) & 1 = (1_r‘𝑌) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) = (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑀))))

Theorem

chpmatply1 22333

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 = (Poly₁‘𝑅) & 𝐸 = (Base‘𝑃) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) ∈ 𝐸)

Theorem

chpmatval2 22334*

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 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & − = (-_g‘𝑌) & 𝑋 = (var₁‘𝑅) & · = ( ·_𝑠 ‘𝑌) & 𝑇 = (𝑁 matToPolyMat 𝑅) & 1 = (1_r‘𝑌) & 𝐺 = (SymGrp‘𝑁) & 𝐻 = (Base‘𝐺) & 𝑍 = (ℤRHom‘𝑃) & 𝑆 = (pmSgn‘𝑁) & 𝑈 = (mulGrp‘𝑃) & × = (._r‘𝑃) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) = (𝑃 Σ_g (𝑝 ∈ 𝐻 ↦ (((𝑍 ∘ 𝑆)‘𝑝) × (𝑈 Σ_g (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)((𝑋 · 1 ) − (𝑇‘𝑀))𝑥)))))))

Theorem

chpmat0d 22335

The characteristic polynomial of the empty matrix. (Contributed by AV, 6-Aug-2019.)

𝐶 = (∅ CharPlyMat 𝑅) ⇒ (𝑅 ∈ Ring → (𝐶‘∅) = (1_r‘(Poly₁‘𝑅)))

Theorem

chpmat1dlem 22336

Lemma for chpmat1d 22337. (Contributed by AV, 7-Aug-2019.)

𝐶 = (𝑁 CharPlyMat 𝑅) & 𝑃 = (Poly₁‘𝑅) & 𝐴 = (𝑁 Mat 𝑅) & 𝐵 = (Base‘𝐴) & 𝑋 = (var₁‘𝑅) & − = (-_g‘𝑃) & 𝑆 = (algSc‘𝑃) & 𝐺 = (𝑁 Mat 𝑃) & 𝑇 = (𝑁 matToPolyMat 𝑅) ⇒ ((𝑅 ∈ Ring ∧ (𝑁 = {𝐼} ∧ 𝐼 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → (𝐼((𝑋( ·_𝑠 ‘𝐺)(1_r‘𝐺))(-_g‘𝐺)(𝑇‘𝑀))𝐼) = (𝑋 − (𝑆‘(𝐼𝑀𝐼))))

Theorem

chpmat1d 22337

The characteristic polynomial of a matrix with dimension 1. (Contributed by AV, 7-Aug-2019.)

𝐶 = (𝑁 CharPlyMat 𝑅) & 𝑃 = (Poly₁‘𝑅) & 𝐴 = (𝑁 Mat 𝑅) & 𝐵 = (Base‘𝐴) & 𝑋 = (var₁‘𝑅) & − = (-_g‘𝑃) & 𝑆 = (algSc‘𝑃) ⇒ ((𝑅 ∈ CRing ∧ (𝑁 = {𝐼} ∧ 𝐼 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) = (𝑋 − (𝑆‘(𝐼𝑀𝐼))))

Theorem

chpdmatlem0 22338

Lemma 0 for chpdmat 22342. (Contributed by AV, 18-Aug-2019.)

𝐶 = (𝑁 CharPlyMat 𝑅) & 𝑃 = (Poly₁‘𝑅) & 𝐴 = (𝑁 Mat 𝑅) & 𝑆 = (algSc‘𝑃) & 𝐵 = (Base‘𝐴) & 𝑋 = (var₁‘𝑅) & 0 = (0_g‘𝑅) & 𝐺 = (mulGrp‘𝑃) & − = (-_g‘𝑃) & 𝑄 = (𝑁 Mat 𝑃) & 1 = (1_r‘𝑄) & · = ( ·_𝑠 ‘𝑄) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑋 · 1 ) ∈ (Base‘𝑄))

Theorem

chpdmatlem1 22339

Lemma 1 for chpdmat 22342. (Contributed by AV, 18-Aug-2019.)

𝐶 = (𝑁 CharPlyMat 𝑅) & 𝑃 = (Poly₁‘𝑅) & 𝐴 = (𝑁 Mat 𝑅) & 𝑆 = (algSc‘𝑃) & 𝐵 = (Base‘𝐴) & 𝑋 = (var₁‘𝑅) & 0 = (0_g‘𝑅) & 𝐺 = (mulGrp‘𝑃) & − = (-_g‘𝑃) & 𝑄 = (𝑁 Mat 𝑃) & 1 = (1_r‘𝑄) & · = ( ·_𝑠 ‘𝑄) & 𝑍 = (-_g‘𝑄) & 𝑇 = (𝑁 matToPolyMat 𝑅) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((𝑋 · 1 )𝑍(𝑇‘𝑀)) ∈ (Base‘𝑄))

Theorem

chpdmatlem2 22340

Lemma 2 for chpdmat 22342. (Contributed by AV, 18-Aug-2019.)

𝐶 = (𝑁 CharPlyMat 𝑅) & 𝑃 = (Poly₁‘𝑅) & 𝐴 = (𝑁 Mat 𝑅) & 𝑆 = (algSc‘𝑃) & 𝐵 = (Base‘𝐴) & 𝑋 = (var₁‘𝑅) & 0 = (0_g‘𝑅) & 𝐺 = (mulGrp‘𝑃) & − = (-_g‘𝑃) & 𝑄 = (𝑁 Mat 𝑃) & 1 = (1_r‘𝑄) & · = ( ·_𝑠 ‘𝑄) & 𝑍 = (-_g‘𝑄) & 𝑇 = (𝑁 matToPolyMat 𝑅) ⇒ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) ∧ 𝑖 ≠ 𝑗) ∧ (𝑖𝑀𝑗) = 0 ) → (𝑖((𝑋 · 1 )𝑍(𝑇‘𝑀))𝑗) = (0_g‘𝑃))

Theorem

chpdmatlem3 22341

Lemma 3 for chpdmat 22342. (Contributed by AV, 18-Aug-2019.)

𝐶 = (𝑁 CharPlyMat 𝑅) & 𝑃 = (Poly₁‘𝑅) & 𝐴 = (𝑁 Mat 𝑅) & 𝑆 = (algSc‘𝑃) & 𝐵 = (Base‘𝐴) & 𝑋 = (var₁‘𝑅) & 0 = (0_g‘𝑅) & 𝐺 = (mulGrp‘𝑃) & − = (-_g‘𝑃) & 𝑄 = (𝑁 Mat 𝑃) & 1 = (1_r‘𝑄) & · = ( ·_𝑠 ‘𝑄) & 𝑍 = (-_g‘𝑄) & 𝑇 = (𝑁 matToPolyMat 𝑅) ⇒ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝐾 ∈ 𝑁) → (𝐾((𝑋 · 1 )𝑍(𝑇‘𝑀))𝐾) = (𝑋 − (𝑆‘(𝐾𝑀𝐾))))

Theorem

chpdmat 22342*

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 = (Poly₁‘𝑅) & 𝐴 = (𝑁 Mat 𝑅) & 𝑆 = (algSc‘𝑃) & 𝐵 = (Base‘𝐴) & 𝑋 = (var₁‘𝑅) & 0 = (0_g‘𝑅) & 𝐺 = (mulGrp‘𝑃) & − = (-_g‘𝑃) ⇒ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) → (𝐶‘𝑀) = (𝐺 Σ_g (𝑘 ∈ 𝑁 ↦ (𝑋 − (𝑆‘(𝑘𝑀𝑘))))))

Theorem

chpscmat 22343*

The characteristic polynomial of a (nonempty!) scalar matrix. (Contributed by AV, 21-Aug-2019.)

Theorem

chpscmat0 22344*

The characteristic polynomial of a (nonempty!) scalar matrix, expressed with its diagonal element. (Contributed by AV, 21-Aug-2019.)

𝐶 = (𝑁 CharPlyMat 𝑅) & 𝑃 = (Poly₁‘𝑅) & 𝐴 = (𝑁 Mat 𝑅) & 𝑋 = (var₁‘𝑅) & 𝐺 = (mulGrp‘𝑃) & ↑ = (._g‘𝐺) & 𝐷 = {𝑚 ∈ (Base‘𝐴) ∣ ∃𝑐 ∈ (Base‘𝑅)∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0_g‘𝑅))} & 𝑆 = (algSc‘𝑃) & − = (-_g‘𝑃) ⇒ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = (𝐼𝑀𝐼))) → (𝐶‘𝑀) = ((♯‘𝑁) ↑ (𝑋 − (𝑆‘(𝐼𝑀𝐼)))))

Theorem

chpscmatgsumbin 22345*

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 = (Poly₁‘𝑅) & 𝐴 = (𝑁 Mat 𝑅) & 𝑋 = (var₁‘𝑅) & 𝐺 = (mulGrp‘𝑃) & ↑ = (._g‘𝐺) & 𝐷 = {𝑚 ∈ (Base‘𝐴) ∣ ∃𝑐 ∈ (Base‘𝑅)∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0_g‘𝑅))} & 𝑆 = (algSc‘𝑃) & − = (-_g‘𝑃) & 𝐹 = (._g‘𝑃) & 𝐻 = (mulGrp‘𝑅) & 𝐸 = (._g‘𝐻) & 𝐼 = (inv_g‘𝑅) & · = ( ·_𝑠 ‘𝑃) ⇒ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐽 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = (𝐽𝑀𝐽))) → (𝐶‘𝑀) = (𝑃 Σ_g (𝑙 ∈ (0...(♯‘𝑁)) ↦ (((♯‘𝑁)C𝑙)𝐹((((♯‘𝑁) − 𝑙)𝐸(𝐼‘(𝐽𝑀𝐽))) · (𝑙 ↑ 𝑋))))))

Theorem

chpscmatgsummon 22346*

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 = (Poly₁‘𝑅) & 𝐴 = (𝑁 Mat 𝑅) & 𝑋 = (var₁‘𝑅) & 𝐺 = (mulGrp‘𝑃) & ↑ = (._g‘𝐺) & 𝐷 = {𝑚 ∈ (Base‘𝐴) ∣ ∃𝑐 ∈ (Base‘𝑅)∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0_g‘𝑅))} & 𝑆 = (algSc‘𝑃) & − = (-_g‘𝑃) & 𝐹 = (._g‘𝑃) & 𝐻 = (mulGrp‘𝑅) & 𝐸 = (._g‘𝐻) & 𝐼 = (inv_g‘𝑅) & · = ( ·_𝑠 ‘𝑃) & 𝑍 = (._g‘𝑅) ⇒ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐽 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = (𝐽𝑀𝐽))) → (𝐶‘𝑀) = (𝑃 Σ_g (𝑙 ∈ (0...(♯‘𝑁)) ↦ ((((♯‘𝑁)C𝑙)𝑍(((♯‘𝑁) − 𝑙)𝐸(𝐼‘(𝐽𝑀𝐽)))) · (𝑙 ↑ 𝑋)))))

Theorem

chp0mat 22347

The characteristic polynomial of the zero matrix. (Contributed by AV, 18-Aug-2019.)

𝐶 = (𝑁 CharPlyMat 𝑅) & 𝑃 = (Poly₁‘𝑅) & 𝐴 = (𝑁 Mat 𝑅) & 𝑋 = (var₁‘𝑅) & 𝐺 = (mulGrp‘𝑃) & ↑ = (._g‘𝐺) & 0 = (0_g‘𝐴) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝐶‘ 0 ) = ((♯‘𝑁) ↑ 𝑋))

Theorem

chpidmat 22348

The characteristic polynomial of the identity matrix. (Contributed by AV, 19-Aug-2019.)

𝐶 = (𝑁 CharPlyMat 𝑅) & 𝑃 = (Poly₁‘𝑅) & 𝐴 = (𝑁 Mat 𝑅) & 𝑋 = (var₁‘𝑅) & 𝐺 = (mulGrp‘𝑃) & ↑ = (._g‘𝐺) & 𝐼 = (1_r‘𝐴) & 𝑆 = (algSc‘𝑃) & 1 = (1_r‘𝑅) & − = (-_g‘𝑃) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝐶‘𝐼) = ((♯‘𝑁) ↑ (𝑋 − (𝑆‘ 1 ))))

Theorem

chmaidscmat 22349

The characteristic polynomial of a matrix multiplied with the identity matrix is a scalar matrix. (Contributed by AV, 30-Oct-2019.) (Revised by AV, 5-Jul-2022.)

𝐴 = (𝑁 Mat 𝑅) & 𝐵 = (Base‘𝐴) & 𝐶 = (𝑁 CharPlyMat 𝑅) & 𝑃 = (Poly₁‘𝑅) & 𝐸 = (Base‘𝑃) & 𝑌 = (𝑁 Mat 𝑃) & 𝐾 = (Base‘𝑌) & · = ( ·_𝑠 ‘𝑌) & 1 = (1_r‘𝑌) & 𝑆 = (𝑁 ScMat 𝑃) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝐶‘𝑀) · 1 ) ∈ 𝑆)

11.7.2 The characteristic factor function G

In this subsection the function 𝐺 = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) 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 22379. 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 22367, cayhamlem3 22388 and cayhamlem4 22389.

Theorem

fvmptnn04if 22350*

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)))) & (𝜑 → 𝑆 ∈ ℕ) & (𝜑 → 𝑁 ∈ ℕ₀) & (𝜑 → 𝑌 ∈ 𝑉) & ((𝜑 ∧ 𝑁 = 0) → 𝑌 = ⦋𝑁 / 𝑛⦌𝐴) & ((𝜑 ∧ 0 < 𝑁 ∧ 𝑁 < 𝑆) → 𝑌 = ⦋𝑁 / 𝑛⦌𝐵) & ((𝜑 ∧ 𝑁 = 𝑆) → 𝑌 = ⦋𝑁 / 𝑛⦌𝐶) & ((𝜑 ∧ 𝑆 < 𝑁) → 𝑌 = ⦋𝑁 / 𝑛⦌𝐷) ⇒ (𝜑 → (𝐺‘𝑁) = 𝑌)

Theorem

fvmptnn04ifa 22351*

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)))) & (𝜑 → 𝑆 ∈ ℕ) & (𝜑 → 𝑁 ∈ ℕ₀) ⇒ ((𝜑 ∧ 𝑁 = 0 ∧ ⦋𝑁 / 𝑛⦌𝐴 ∈ 𝑉) → (𝐺‘𝑁) = ⦋𝑁 / 𝑛⦌𝐴)

Theorem

fvmptnn04ifb 22352*

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)))) & (𝜑 → 𝑆 ∈ ℕ) & (𝜑 → 𝑁 ∈ ℕ₀) ⇒ ((𝜑 ∧ (0 < 𝑁 ∧ 𝑁 < 𝑆) ∧ ⦋𝑁 / 𝑛⦌𝐵 ∈ 𝑉) → (𝐺‘𝑁) = ⦋𝑁 / 𝑛⦌𝐵)

Theorem

fvmptnn04ifc 22353*

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)))) & (𝜑 → 𝑆 ∈ ℕ) & (𝜑 → 𝑁 ∈ ℕ₀) ⇒ ((𝜑 ∧ 𝑁 = 𝑆 ∧ ⦋𝑁 / 𝑛⦌𝐶 ∈ 𝑉) → (𝐺‘𝑁) = ⦋𝑁 / 𝑛⦌𝐶)

Theorem

fvmptnn04ifd 22354*

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)))) & (𝜑 → 𝑆 ∈ ℕ) & (𝜑 → 𝑁 ∈ ℕ₀) ⇒ ((𝜑 ∧ 𝑆 < 𝑁 ∧ ⦋𝑁 / 𝑛⦌𝐷 ∈ 𝑉) → (𝐺‘𝑁) = ⦋𝑁 / 𝑛⦌𝐷)

Theorem

chfacfisf 22355*

The "characteristic factor function" is a function from the nonnegative integers to polynomial matrices. (Contributed by AV, 8-Nov-2019.)

Theorem

chfacfisfcpmat 22356*

The "characteristic factor function" is a function from the nonnegative integers to constant polynomial matrices. (Contributed by AV, 19-Nov-2019.)

Theorem

chfacffsupp 22357*

The "characteristic factor function" is finitely supported. (Contributed by AV, 20-Nov-2019.) (Proof shortened by AV, 23-Dec-2019.)

Theorem

chfacfscmulcl 22358*

Closure of a scaled value of the "characteristic factor function". (Contributed by AV, 9-Nov-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝐵 = (Base‘𝐴) & 𝑃 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & × = (._r‘𝑌) & − = (-_g‘𝑌) & 0 = (0_g‘𝑌) & 𝑇 = (𝑁 matToPolyMat 𝑅) & 𝐺 = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) & 𝑋 = (var₁‘𝑅) & · = ( ·_𝑠 ‘𝑌) & ↑ = (._g‘(mulGrp‘𝑃)) ⇒ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝐾 ∈ ℕ₀) → ((𝐾 ↑ 𝑋) · (𝐺‘𝐾)) ∈ (Base‘𝑌))

Theorem

chfacfscmul0 22359*

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 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & × = (._r‘𝑌) & − = (-_g‘𝑌) & 0 = (0_g‘𝑌) & 𝑇 = (𝑁 matToPolyMat 𝑅) & 𝐺 = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) & 𝑋 = (var₁‘𝑅) & · = ( ·_𝑠 ‘𝑌) & ↑ = (._g‘(mulGrp‘𝑃)) ⇒ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝐾 ∈ (ℤ_≥‘(𝑠 + 2))) → ((𝐾 ↑ 𝑋) · (𝐺‘𝐾)) = 0 )

Theorem

chfacfscmulfsupp 22360*

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 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & × = (._r‘𝑌) & − = (-_g‘𝑌) & 0 = (0_g‘𝑌) & 𝑇 = (𝑁 matToPolyMat 𝑅) & 𝐺 = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) & 𝑋 = (var₁‘𝑅) & · = ( ·_𝑠 ‘𝑌) & ↑ = (._g‘(mulGrp‘𝑃)) ⇒ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑖 ∈ ℕ₀ ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖))) finSupp 0 )

Theorem

chfacfscmulgsum 22361*

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 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & × = (._r‘𝑌) & − = (-_g‘𝑌) & 0 = (0_g‘𝑌) & 𝑇 = (𝑁 matToPolyMat 𝑅) & 𝐺 = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) & 𝑋 = (var₁‘𝑅) & · = ( ·_𝑠 ‘𝑌) & ↑ = (._g‘(mulGrp‘𝑃)) & + = (+_g‘𝑌) ⇒ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ ℕ₀ ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))

Theorem

chfacfpmmulcl 22362*

Closure of the value of the "characteristic factor function" multiplied with a constant polynomial matrix. (Contributed by AV, 23-Nov-2019.)

Theorem

chfacfpmmul0 22363*

The value of the "characteristic factor function" multiplied with a constant polynomial matrix is zero almost always. (Contributed by AV, 23-Nov-2019.)

Theorem

chfacfpmmulfsupp 22364*

A mapping of values of the "characteristic factor function" multiplied with a constant polynomial matrix is finitely supported. (Contributed by AV, 23-Nov-2019.)

Theorem

chfacfpmmulgsum 22365*

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 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & × = (._r‘𝑌) & − = (-_g‘𝑌) & 0 = (0_g‘𝑌) & 𝑇 = (𝑁 matToPolyMat 𝑅) & 𝐺 = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) & ↑ = (._g‘(mulGrp‘𝑌)) & + = (+_g‘𝑌) ⇒ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ ℕ₀ ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ (𝑇‘𝑀)) × ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))

Theorem

chfacfpmmulgsum2 22366*

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 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & × = (._r‘𝑌) & − = (-_g‘𝑌) & 0 = (0_g‘𝑌) & 𝑇 = (𝑁 matToPolyMat 𝑅) & 𝐺 = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) & ↑ = (._g‘(mulGrp‘𝑌)) & + = (+_g‘𝑌) ⇒ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ ℕ₀ ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ (((𝑖 ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) − (((𝑖 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑖)))))) + ((((𝑠 + 1) ↑ (𝑇‘𝑀)) × (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))

Theorem

cayhamlem1 22367*

Lemma 1 for cayleyhamilton 22391. (Contributed by AV, 11-Nov-2019.)

11.7.3 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 * I_n - 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 * I_n - A) .

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

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

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 B_i of numbers, such that one has

(3) B = sum_{i = 0 to (n-1)} t^i B_i .

(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) * I_n = ( t * I_n - A ) x B
= ( t * I_n - A ) x sum_{i = 0 to (n-1)} t^i * B_i = sum_{i = 0 to (n-1)} t * I_n x t^i B_i - sum_{i = 0 to (n-1)} A * t^i B_i = sum_{i = 0 to (n-1)} t^(i+1) * B_i - sum_{i = 0 to (n-1)} t^i * A x B_i = t^n B_n-1 + sum_{i = 1 to (n-1)} t^i * ( B_i-1 - A x B_i ) - A x B₀ .

Writing

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

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) B_n-1 = I_n , B_i-1 - A x B_i = c_i * I_n for 1 <= i <= n-1 , - A x B₀ = c₀ * I_n .

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

(7) A^n B_n-1 + sum_{i
= 1 to (n-1)} ( A^i x B_i-1 - A^(i+1) x B_i ) - A x B₀ = A^n + c_n-1 * A^(n-1) + ... + c₁ * A + c₀ * I_n .

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 I_n is the n x n identity matrix, then the characteristic polynomial of A is defined as p(t) = det(t * I_n - 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 * I_n - 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 22328, the characteristic polynomial of an 𝑁 x 𝑁 matrix 𝑀 over a ring 𝑅 is defined as

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

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 = (var₁‘𝑅) 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 (𝑇‘𝑀) = ((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 22332).

By this definition, it is ensured that ((𝑋 · 1 ) − (𝑇‘𝑀)), the matrix whose determinant 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 22332). 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 ) − (𝑇‘𝑀)) 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 21819), also the characteristic polynomial can be written in this way:

p(M) = sum_i ( p_i * 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( p_i * M^i)", see also cayleyhamilton1 22393. 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 "A_B") 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	(Poly₁‘𝑅)	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, X_R	(var₁‘𝑅)	The variable of polynomials over (the ring) 𝑅
𝑌, X_A	(var₁‘𝐴)	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
(𝑈‘𝑝), 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	(Poly₁‘𝐴)	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
+, +	(+_g‘𝑌)	The addition of polynomial matrices
−, -	(-_g‘𝑌)	The subtraction of polynomial matrices
·, *_Y	( ·_𝑠 ‘𝑌)	The multiplication of a polynomial matrix with a scalar ( i. e. a polynomial)
*_A	( ·_𝑠 ‘𝐴)	The multiplication of a matrix with a scalar ( i. e. an element of the underlying ring)
*_Q	( ·_𝑠 ‘𝑄)	The multiplication of a polynomial over matrices with a scalar ( i. e. a matrix)
×, x_Y	(._r‘𝑌)	The multiplication of polynomial matrices
x_A	(._r‘𝐴)	The multiplication of matrices
x_Q	(._r‘𝑄)	The multiplication of polynomials over matrices
1, 1_Y	(1_r‘𝑌)	The identity matrix in the algebra of polynomial matrices over 𝑅
1_A	(1_r‘𝐴)	The identity matrix in the algebra of matrices over 𝑅
0, 0_Y	(0_g‘𝑌)	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)	(𝑇‘𝑀)	The matrix M transformed into a polynomial 𝑁 x 𝑁 matrix over 𝑅
U(M)	(𝑈‘𝑀)	The (constant) polynomial 𝑁 x 𝑁 matrix M transformed into a matrix over the ring 𝑅. Inverse function of 𝑇: (𝑇‘(𝑈‘𝑀)) = 𝑀 (see m2cpminvid2 22256 )
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 ) − (𝑇‘𝑀))	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)	(𝐶‘𝑀)	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))	(𝐽‘𝐼)	The adjugate of the characteristic matrix of the matrix 𝑀

Using this notation, we have:

"c(M) e. B(Y)", or 𝐼 ∈ (Base‘𝑌), see chmatcl 22329
"p(M) e. B(P)", or 𝐾 ∈ (Base‘𝑃), see chpmatply1 22333
"T(M) e. B(Y)", or (𝑇‘𝑀) ∈ (Base‘𝑌), see mat2pmatbas 22227
𝐽:(Base‘𝑌)⟶(Base‘𝑌), see maduf 22142
"adj(cm(M)) e. B(Y)", or 𝑊 ∈ (Base‘𝑌)

Following the proof shown in Wikipedia, the following steps are performed:

Write 𝑊, the adjugate of the characteristic matrix, as a finite sum of scaled monomials, see pmatcollpw3fi1 22289:
adj(cm(M)) = sum_{i=0 to s} ( X_R ^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.
Write (𝐼 × 𝑊), the product of the characteristic matrix and its adjugate as finite sum of scaled monomials, see cpmadugsumfi 22378. 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)) = sum_{i=0 to s} ( X_R ^i *_Y ( T₁(b(i-1)) - T₁(M) x_Y T₁(b(i)) ) ) + X_R ^(s+1) *_Y ( T₁(b(s)) - T₁(M) x_Y 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 sum_{i=0 to s} ( X_R ^i *_Y T₁(b(i)) ) [see pmatcollpw3fi1 22289 (step 1.)]
= ( ( X_A *_Y 1_Y ) - T₁(M) ) *_Y sum_{i=0 to s} ( X_R ^i *_Y T₁(b(i)) ) [def. of cm(M)]
= ( X_A *_Y 1_Y ) *_Y sum_{i=0 to s} ( X_R ^i *_Y T₁(b(i)) ) - T₁(M) *_Y sum_{i=0 to s} ( X_R ^i *_Y T₁(b(i)) ) [see ringsubdir 20119]
= sum_{i=0 to s} ( X_R ^i *_Y ( T₁(b(i-1)) - T₁(M) x_Y T₁(b(i)) ) ) + X_R ^(s+1) *_Y ( T₁(b(s)) - T₁(M) x_Y T₁(b(0)) [see cpmadugsumlemF 22377]
This step corresponds partially to (4) in Wikipedia.
Write (𝐼 × 𝑊) as infinite, but finitely supported sum of scaled monomials, see cpmadugsum 22379:
cm(M) * adj(cm(M)) = sum_i ( X_R ^i *_Y G(i) )
This representation is obtained by defining a function G for the coefficients, which we call "characteristic factor function", see chfacfisf 22355, which covers the special terms and the padding with 0. G(i) is a constant polynomial matrix (see chfacfisfcpmat 22356). This step corresponds partially to (4) in Wikipedia, with summands of value 0 added.
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 22369:
p(m) *_Y I_Y = sum_i ( X_R ^i *_Y ( S(p_i) *_Y I_Y ) )
The proof of cpmidgsum 22369 is making use of pmatcollpwscmat 22292, because 𝐻 = (𝐾 · 1 ) is a scalar/diagonal polynomial matrix with the characteristic polynomial "p(M)" as diagonal entries (since p_i is an element of the ring 𝑅, S(p_i) is a (constant) polynomial). This corresponds to (5) in Wikipedia, with summands of value 0 added.
Transform the sum representation of (𝐼 × 𝑊) from step 3. into polynomials over matrices:
T₂(cm(M) * adj(cm(M))) = sum_i ( U(G(i)) *_Q X_A ^i ) [see cpmadumatpoly 22384]
where U(G(i)) is a matrix over the ring 𝑅.
Transform the sum representation of 𝐻 from step 4. into polynomials over matrices:
T₂(p(m) *_Y I_Y) = sum_i ( p_i *_A I_A ) *_Q X_A ^i ) [see cpmidpmat 22374]
Equate the sum representations resulting from steps 5. and 6. by using cpmadurid 22368 to obtain the equation
sum_i ( U(G(i)) *_Q X_A ^i ) = sum_i ( p_i *_A I_A ) *_Q X_A ^i ):
sum_i ( U(G(i)) *_Q X_A ^i )
= T₂(cm(M) * adj(cm(M))) [see step 5.]
= T₂(p(m) *_Y I_Y) [see cpmadurid 22368]
= sum_i ( p_i *_A I_A ) *_Q X_A ^i ) [see step 6.]
Note that this step is contained in the proof of chcoeffeq 22387, see step 9. This step corresponds to the conclusion from (4) and (5) in Wikipedia, with summands of value 0 added.
Compare the sum representations of step 7. to obtain the equations U(G(i)) = p_i *_A I_A , see chcoeffeqlem 22386. 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.
Multiply the equations of the coefficients from step 8. from the left by M^i, and sum up, see chcoeffeq 22387:
sum_i ( M^i x_A U(G(i)) ) = sum_i ( M^i x_A ( p_i *_A I_A) )
This corresponds to (7) in Wikipedia.
Transform the right hand side of the equation in step 9. into an appropriate form, see cayhamlem3 22388:
sum_i ( p_i *_A M^i )
= sum_i ( M^i x_A ( p_i *_A I_A) ) [see cayhamlem2 22385]
= sum_i ( M^i x_A U(G(i)) ) [see chcoeffeq 22387]
Apply the theorem for telescoping sums, see telgsumfz 19857, to the sum sum_i ( T₁(M)^i x_Y G(i) ), which results in an equation to 0:
sum_i ( T₁(M)^i x_Y G(i) ) = 0_Y, see cayhamlem1 22367:
sum_i ( T₁(M)^i x_Y G(i) )
= sum_{i=1 to s} ( T₁(M)^i x_Y T₁(b(i-1)) - T₁(M)^(i+1) x_Y T₁(b(i)) )
+ ( T₁(M)^(s+1) x_Y T₁(b(s)) - T₁(M) x_Y T₁(b(0)) ) [see chfacfpmmulgsum2 22366]
= ( T₁(M) x_Y T₁(b(0)) - T₁(M)^(s+1) x_Y T₁(b(s)) ) + ( T₁ M)^(s+1) x_Y T₁(b(s)) - T₁(M) x_Y T₁(b(0)) ) [see telgsumfz 19857]
= 0_Y [see grpnpncan0 18918] This step corresponds partially to (8) in Wikipedia.
Since 𝑇 is a ring homomorphism (see mat2pmatrhm 22235), 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 22389:
sum_i ( p_i *_A M^i )
= sum_i ( M^i x_A U(G(i)) ) [see cayhamlem3 22388 (step 10.)]
= U(T₁(sum_i ( M^i x_A U(G(i)) ))) [see m2cpminvid 22254]
= U(sum_i T₁( M^i x_A U(G(i)) )) [see gsummptmhm 19807]
= U(sum_i ( T₁(M^i) x_Y T₁(U(G(i))) )) [see rhmmul 20263]
= U(sum_i ( T₁(M)^i x_Y T₁(U(G(i))) )) [see mhmmulg 18994]
= U(sum_i ( T₁(M)^i x_Y G(i) )) [see m2cpminvid2 22256 ]
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 22230 and mat2pmatrhm 22235) to transform the equality resulting from steps 11. and 12. into the desired equation sum_i ( p_i *_A M^i ) = 0_A , see cayleyhamilton 22391 resp. cayleyhamilton0 22390:
sum_i ( p_i *_A M^i )
= U(sum_i ( T₁(M)^i x_Y G(i) )) [see cayhamlem4 22389 (step 12.)]
= U(0_Y ) [see cayhamlem1 22367 (step 11.)]
= 0_A [see m2cpminv0 22262]

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!

Theorem

cpmadurid 22368

The right-hand fundamental relation of the adjugate (see madurid 22145) 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 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & 𝑋 = (var₁‘𝑅) & 𝑇 = (𝑁 matToPolyMat 𝑅) & − = (-_g‘𝑌) & · = ( ·_𝑠 ‘𝑌) & 1 = (1_r‘𝑌) & 𝐼 = ((𝑋 · 1 ) − (𝑇‘𝑀)) & 𝐽 = (𝑁 maAdju 𝑃) & × = (._r‘𝑌) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐼 × (𝐽‘𝐼)) = ((𝐶‘𝑀) · 1 ))

Theorem

cpmidgsum 22369*

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 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & 𝑋 = (var₁‘𝑅) & ↑ = (._g‘(mulGrp‘𝑃)) & · = ( ·_𝑠 ‘𝑌) & 1 = (1_r‘𝑌) & 𝑈 = (algSc‘𝑃) & 𝐶 = (𝑁 CharPlyMat 𝑅) & 𝐾 = (𝐶‘𝑀) & 𝐻 = (𝐾 · 1 ) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐻 = (𝑌 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛 ↑ 𝑋) · ((𝑈‘((coe₁‘𝐾)‘𝑛)) · 1 )))))

Theorem

cpmidgsumm2pm 22370*

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 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & 𝑋 = (var₁‘𝑅) & ↑ = (._g‘(mulGrp‘𝑃)) & · = ( ·_𝑠 ‘𝑌) & 1 = (1_r‘𝑌) & 𝑈 = (algSc‘𝑃) & 𝐶 = (𝑁 CharPlyMat 𝑅) & 𝐾 = (𝐶‘𝑀) & 𝐻 = (𝐾 · 1 ) & 𝑂 = (1_r‘𝐴) & ∗ = ( ·_𝑠 ‘𝐴) & 𝑇 = (𝑁 matToPolyMat 𝑅) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐻 = (𝑌 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛 ↑ 𝑋) · (𝑇‘(((coe₁‘𝐾)‘𝑛) ∗ 𝑂))))))

Theorem

cpmidpmatlem1 22371*

Lemma 1 for cpmidpmat 22374. (Contributed by AV, 13-Nov-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝐵 = (Base‘𝐴) & 𝑃 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & 𝑋 = (var₁‘𝑅) & ↑ = (._g‘(mulGrp‘𝑃)) & · = ( ·_𝑠 ‘𝑌) & 1 = (1_r‘𝑌) & 𝑈 = (algSc‘𝑃) & 𝐶 = (𝑁 CharPlyMat 𝑅) & 𝐾 = (𝐶‘𝑀) & 𝐻 = (𝐾 · 1 ) & 𝑂 = (1_r‘𝐴) & ∗ = ( ·_𝑠 ‘𝐴) & 𝑇 = (𝑁 matToPolyMat 𝑅) & 𝐺 = (𝑘 ∈ ℕ₀ ↦ (((coe₁‘𝐾)‘𝑘) ∗ 𝑂)) ⇒ (𝐿 ∈ ℕ₀ → (𝐺‘𝐿) = (((coe₁‘𝐾)‘𝐿) ∗ 𝑂))

Theorem

cpmidpmatlem2 22372*

Lemma 2 for cpmidpmat 22374. (Contributed by AV, 14-Nov-2019.) (Proof shortened by AV, 7-Dec-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝐵 = (Base‘𝐴) & 𝑃 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & 𝑋 = (var₁‘𝑅) & ↑ = (._g‘(mulGrp‘𝑃)) & · = ( ·_𝑠 ‘𝑌) & 1 = (1_r‘𝑌) & 𝑈 = (algSc‘𝑃) & 𝐶 = (𝑁 CharPlyMat 𝑅) & 𝐾 = (𝐶‘𝑀) & 𝐻 = (𝐾 · 1 ) & 𝑂 = (1_r‘𝐴) & ∗ = ( ·_𝑠 ‘𝐴) & 𝑇 = (𝑁 matToPolyMat 𝑅) & 𝐺 = (𝑘 ∈ ℕ₀ ↦ (((coe₁‘𝐾)‘𝑘) ∗ 𝑂)) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐺 ∈ (𝐵 ↑_m ℕ₀))

Theorem

cpmidpmatlem3 22373*

Lemma 3 for cpmidpmat 22374. (Contributed by AV, 14-Nov-2019.) (Proof shortened by AV, 7-Dec-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝐵 = (Base‘𝐴) & 𝑃 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & 𝑋 = (var₁‘𝑅) & ↑ = (._g‘(mulGrp‘𝑃)) & · = ( ·_𝑠 ‘𝑌) & 1 = (1_r‘𝑌) & 𝑈 = (algSc‘𝑃) & 𝐶 = (𝑁 CharPlyMat 𝑅) & 𝐾 = (𝐶‘𝑀) & 𝐻 = (𝐾 · 1 ) & 𝑂 = (1_r‘𝐴) & ∗ = ( ·_𝑠 ‘𝐴) & 𝑇 = (𝑁 matToPolyMat 𝑅) & 𝐺 = (𝑘 ∈ ℕ₀ ↦ (((coe₁‘𝐾)‘𝑘) ∗ 𝑂)) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝐺 finSupp (0_g‘𝐴))

Theorem

cpmidpmat 22374*

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 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & 𝑋 = (var₁‘𝑅) & ↑ = (._g‘(mulGrp‘𝑃)) & · = ( ·_𝑠 ‘𝑌) & 1 = (1_r‘𝑌) & 𝑈 = (algSc‘𝑃) & 𝐶 = (𝑁 CharPlyMat 𝑅) & 𝐾 = (𝐶‘𝑀) & 𝐻 = (𝐾 · 1 ) & 𝑂 = (1_r‘𝐴) & ∗ = ( ·_𝑠 ‘𝐴) & 𝑇 = (𝑁 matToPolyMat 𝑅) & 𝑊 = (Base‘𝑌) & 𝑄 = (Poly₁‘𝐴) & 𝑍 = (var₁‘𝐴) & ∙ = ( ·_𝑠 ‘𝑄) & 𝐸 = (._g‘(mulGrp‘𝑄)) & 𝐼 = (𝑁 pMatToMatPoly 𝑅) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐼‘𝐻) = (𝑄 Σ_g (𝑛 ∈ ℕ₀ ↦ ((((coe₁‘𝐾)‘𝑛) ∗ 𝑂) ∙ (𝑛𝐸𝑍)))))

Theorem

cpmadugsumlemB 22375*

Lemma B for cpmadugsum 22379. (Contributed by AV, 2-Nov-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝐵 = (Base‘𝐴) & 𝑃 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & 𝑇 = (𝑁 matToPolyMat 𝑅) & 𝑋 = (var₁‘𝑅) & ↑ = (._g‘(mulGrp‘𝑃)) & · = ( ·_𝑠 ‘𝑌) & × = (._r‘𝑌) & 1 = (1_r‘𝑌) ⇒ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑋 · 1 ) × (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))) = (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ (((𝑖 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))))

Theorem

cpmadugsumlemC 22376*

Lemma C for cpmadugsum 22379. (Contributed by AV, 2-Nov-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝐵 = (Base‘𝐴) & 𝑃 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & 𝑇 = (𝑁 matToPolyMat 𝑅) & 𝑋 = (var₁‘𝑅) & ↑ = (._g‘(mulGrp‘𝑃)) & · = ( ·_𝑠 ‘𝑌) & × = (._r‘𝑌) & 1 = (1_r‘𝑌) ⇒ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ₀ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → ((𝑇‘𝑀) × (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))) = (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖)))))))

Theorem

cpmadugsumlemF 22377*

Lemma F for cpmadugsum 22379. (Contributed by AV, 7-Nov-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝐵 = (Base‘𝐴) & 𝑃 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & 𝑇 = (𝑁 matToPolyMat 𝑅) & 𝑋 = (var₁‘𝑅) & ↑ = (._g‘(mulGrp‘𝑃)) & · = ( ·_𝑠 ‘𝑌) & × = (._r‘𝑌) & 1 = (1_r‘𝑌) & + = (+_g‘𝑌) & − = (-_g‘𝑌) ⇒ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (((𝑋 · 1 ) × (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖)))))) − ((𝑇‘𝑀) × (𝑌 Σ_g (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))))) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))

Theorem

cpmadugsumfi 22378*

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 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & 𝑇 = (𝑁 matToPolyMat 𝑅) & 𝑋 = (var₁‘𝑅) & ↑ = (._g‘(mulGrp‘𝑃)) & · = ( ·_𝑠 ‘𝑌) & × = (._r‘𝑌) & 1 = (1_r‘𝑌) & + = (+_g‘𝑌) & − = (-_g‘𝑌) & 𝐼 = ((𝑋 · 1 ) − (𝑇‘𝑀)) & 𝐽 = (𝑁 maAdju 𝑃) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑_m (0...𝑠))(𝐼 × (𝐽‘𝐼)) = ((𝑌 Σ_g (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0))))))

Theorem

cpmadugsum 22379*

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 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & 𝑇 = (𝑁 matToPolyMat 𝑅) & 𝑋 = (var₁‘𝑅) & ↑ = (._g‘(mulGrp‘𝑃)) & · = ( ·_𝑠 ‘𝑌) & × = (._r‘𝑌) & 1 = (1_r‘𝑌) & + = (+_g‘𝑌) & − = (-_g‘𝑌) & 𝐼 = ((𝑋 · 1 ) − (𝑇‘𝑀)) & 𝐽 = (𝑁 maAdju 𝑃) & 0 = (0_g‘𝑌) & 𝐺 = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑_m (0...𝑠))(𝐼 × (𝐽‘𝐼)) = (𝑌 Σ_g (𝑖 ∈ ℕ₀ ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))))

Theorem

cpmidgsum2 22380*

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 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & 𝑇 = (𝑁 matToPolyMat 𝑅) & 𝑋 = (var₁‘𝑅) & ↑ = (._g‘(mulGrp‘𝑃)) & · = ( ·_𝑠 ‘𝑌) & × = (._r‘𝑌) & 1 = (1_r‘𝑌) & + = (+_g‘𝑌) & − = (-_g‘𝑌) & 𝐼 = ((𝑋 · 1 ) − (𝑇‘𝑀)) & 𝐽 = (𝑁 maAdju 𝑃) & 0 = (0_g‘𝑌) & 𝐺 = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) & 𝐶 = (𝑁 CharPlyMat 𝑅) & 𝐾 = (𝐶‘𝑀) & 𝐻 = (𝐾 · 1 ) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑_m (0...𝑠))𝐻 = (𝑌 Σ_g (𝑖 ∈ ℕ₀ ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))))

Theorem

cpmidg2sum 22381*

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 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & 𝑇 = (𝑁 matToPolyMat 𝑅) & 𝑋 = (var₁‘𝑅) & ↑ = (._g‘(mulGrp‘𝑃)) & · = ( ·_𝑠 ‘𝑌) & × = (._r‘𝑌) & 1 = (1_r‘𝑌) & + = (+_g‘𝑌) & − = (-_g‘𝑌) & 𝐼 = ((𝑋 · 1 ) − (𝑇‘𝑀)) & 𝐽 = (𝑁 maAdju 𝑃) & 0 = (0_g‘𝑌) & 𝐺 = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) & 𝐶 = (𝑁 CharPlyMat 𝑅) & 𝐾 = (𝐶‘𝑀) & 𝑈 = (algSc‘𝑃) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑_m (0...𝑠))(𝑌 Σ_g (𝑖 ∈ ℕ₀ ↦ ((𝑖 ↑ 𝑋) · ((𝑈‘((coe₁‘𝐾)‘𝑖)) · 1 )))) = (𝑌 Σ_g (𝑖 ∈ ℕ₀ ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))))

Theorem

cpmadumatpolylem1 22382*

Lemma 1 for cpmadumatpoly 22384. (Contributed by AV, 20-Nov-2019.) (Revised by AV, 15-Dec-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝐵 = (Base‘𝐴) & 𝑃 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & 𝑇 = (𝑁 matToPolyMat 𝑅) & × = (._r‘𝑌) & − = (-_g‘𝑌) & 0 = (0_g‘𝑌) & 𝐺 = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) & 𝑆 = (𝑁 ConstPolyMat 𝑅) & · = ( ·_𝑠 ‘𝑌) & 1 = (1_r‘𝑌) & 𝑍 = (var₁‘𝑅) & 𝐷 = ((𝑍 · 1 ) − (𝑇‘𝑀)) & 𝐽 = (𝑁 maAdju 𝑃) & 𝑊 = (Base‘𝑌) & 𝑄 = (Poly₁‘𝐴) & 𝑋 = (var₁‘𝐴) & ∗ = ( ·_𝑠 ‘𝑄) & ↑ = (._g‘(mulGrp‘𝑄)) & 𝑈 = (𝑁 cPolyMatToMat 𝑅) ⇒ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑠 ∈ ℕ) ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) → (𝑈 ∘ 𝐺) ∈ (𝐵 ↑_m ℕ₀))

Theorem

cpmadumatpolylem2 22383*

Lemma 2 for cpmadumatpoly 22384. (Contributed by AV, 20-Nov-2019.) (Revised by AV, 15-Dec-2019.)

Theorem

cpmadumatpoly 22384*

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 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & 𝑇 = (𝑁 matToPolyMat 𝑅) & × = (._r‘𝑌) & − = (-_g‘𝑌) & 0 = (0_g‘𝑌) & 𝐺 = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) & 𝑆 = (𝑁 ConstPolyMat 𝑅) & · = ( ·_𝑠 ‘𝑌) & 1 = (1_r‘𝑌) & 𝑍 = (var₁‘𝑅) & 𝐷 = ((𝑍 · 1 ) − (𝑇‘𝑀)) & 𝐽 = (𝑁 maAdju 𝑃) & 𝑊 = (Base‘𝑌) & 𝑄 = (Poly₁‘𝐴) & 𝑋 = (var₁‘𝐴) & ∗ = ( ·_𝑠 ‘𝑄) & ↑ = (._g‘(mulGrp‘𝑄)) & 𝑈 = (𝑁 cPolyMatToMat 𝑅) & 𝐼 = (𝑁 pMatToMatPoly 𝑅) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑_m (0...𝑠))(𝐼‘(𝐷 × (𝐽‘𝐷))) = (𝑄 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑈‘(𝐺‘𝑛)) ∗ (𝑛 ↑ 𝑋)))))

Theorem

cayhamlem2 22385

Lemma for cayhamlem3 22388. (Contributed by AV, 24-Nov-2019.)

𝐾 = (Base‘𝑅) & 𝐴 = (𝑁 Mat 𝑅) & 𝐵 = (Base‘𝐴) & 1 = (1_r‘𝐴) & ∗ = ( ·_𝑠 ‘𝐴) & ↑ = (._g‘(mulGrp‘𝐴)) & · = (._r‘𝐴) ⇒ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐻 ∈ (𝐾 ↑_m ℕ₀) ∧ 𝐿 ∈ ℕ₀)) → ((𝐻‘𝐿) ∗ (𝐿 ↑ 𝑀)) = ((𝐿 ↑ 𝑀) · ((𝐻‘𝐿) ∗ 1 )))

Theorem

chcoeffeqlem 22386*

Lemma for chcoeffeq 22387. (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 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & × = (._r‘𝑌) & − = (-_g‘𝑌) & 0 = (0_g‘𝑌) & 𝑇 = (𝑁 matToPolyMat 𝑅) & 𝐶 = (𝑁 CharPlyMat 𝑅) & 𝐾 = (𝐶‘𝑀) & 𝐺 = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) & 𝑊 = (Base‘𝑌) & 1 = (1_r‘𝐴) & ∗ = ( ·_𝑠 ‘𝐴) & 𝑈 = (𝑁 cPolyMatToMat 𝑅) ⇒ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (((Poly₁‘𝐴) Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑈‘(𝐺‘𝑛))( ·_𝑠 ‘(Poly₁‘𝐴))(𝑛(._g‘(mulGrp‘(Poly₁‘𝐴)))(var₁‘𝐴))))) = ((Poly₁‘𝐴) Σ_g (𝑛 ∈ ℕ₀ ↦ ((((coe₁‘𝐾)‘𝑛) ∗ 1 )( ·_𝑠 ‘(Poly₁‘𝐴))(𝑛(._g‘(mulGrp‘(Poly₁‘𝐴)))(var₁‘𝐴))))) → ∀𝑛 ∈ ℕ₀ (𝑈‘(𝐺‘𝑛)) = (((coe₁‘𝐾)‘𝑛) ∗ 1 )))

Theorem

chcoeffeq 22387*

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.)

Theorem

cayhamlem3 22388*

Lemma for cayhamlem4 22389. (Contributed by AV, 24-Nov-2019.) (Revised by AV, 15-Dec-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝐵 = (Base‘𝐴) & 𝑃 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & × = (._r‘𝑌) & − = (-_g‘𝑌) & 0 = (0_g‘𝑌) & 𝑇 = (𝑁 matToPolyMat 𝑅) & 𝐶 = (𝑁 CharPlyMat 𝑅) & 𝐾 = (𝐶‘𝑀) & 𝐺 = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) & 𝑊 = (Base‘𝑌) & 1 = (1_r‘𝐴) & ∗ = ( ·_𝑠 ‘𝐴) & 𝑈 = (𝑁 cPolyMatToMat 𝑅) & ↑ = (._g‘(mulGrp‘𝐴)) & · = (._r‘𝐴) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑_m (0...𝑠))(𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ (((coe₁‘𝐾)‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛 ↑ 𝑀) · (𝑈‘(𝐺‘𝑛))))))

Theorem

cayhamlem4 22389*

Lemma for cayleyhamilton 22391. (Contributed by AV, 25-Nov-2019.) (Revised by AV, 15-Dec-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝐵 = (Base‘𝐴) & 𝑃 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & × = (._r‘𝑌) & − = (-_g‘𝑌) & 0 = (0_g‘𝑌) & 𝑇 = (𝑁 matToPolyMat 𝑅) & 𝐶 = (𝑁 CharPlyMat 𝑅) & 𝐾 = (𝐶‘𝑀) & 𝐺 = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) & 𝑊 = (Base‘𝑌) & 1 = (1_r‘𝐴) & ∗ = ( ·_𝑠 ‘𝐴) & 𝑈 = (𝑁 cPolyMatToMat 𝑅) & ↑ = (._g‘(mulGrp‘𝐴)) & 𝐸 = (._g‘(mulGrp‘𝑌)) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑏 ∈ (𝐵 ↑_m (0...𝑠))(𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ (((coe₁‘𝐾)‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = (𝑈‘(𝑌 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝑛𝐸(𝑇‘𝑀)) × (𝐺‘𝑛))))))

Theorem

cayleyhamilton0 22390*

The Cayley-Hamilton theorem: A matrix over a commutative ring "satisfies its own characteristic equation". This version of cayleyhamilton 22391 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 22392)! (Contributed by AV, 25-Nov-2019.) (Revised by AV, 15-Dec-2019.)

𝐴 = (𝑁 Mat 𝑅) & 𝐵 = (Base‘𝐴) & 0 = (0_g‘𝐴) & 1 = (1_r‘𝐴) & ∗ = ( ·_𝑠 ‘𝐴) & ↑ = (._g‘(mulGrp‘𝐴)) & 𝐶 = (𝑁 CharPlyMat 𝑅) & 𝐾 = (coe₁‘(𝐶‘𝑀)) & 𝑃 = (Poly₁‘𝑅) & 𝑌 = (𝑁 Mat 𝑃) & × = (._r‘𝑌) & − = (-_g‘𝑌) & 𝑍 = (0_g‘𝑌) & 𝑊 = (Base‘𝑌) & 𝐸 = (._g‘(mulGrp‘𝑌)) & 𝑇 = (𝑁 matToPolyMat 𝑅) & 𝐺 = (𝑛 ∈ ℕ₀ ↦ if(𝑛 = 0, (𝑍 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 𝑍, ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) & 𝑈 = (𝑁 cPolyMatToMat 𝑅) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = 0 )

Theorem

cayleyhamilton 22391*

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 = (0_g‘𝐴) & 𝐶 = (𝑁 CharPlyMat 𝑅) & 𝐾 = (coe₁‘(𝐶‘𝑀)) & ∗ = ( ·_𝑠 ‘𝐴) & ↑ = (._g‘(mulGrp‘𝐴)) ⇒ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝐾‘𝑛) ∗ (𝑛 ↑ 𝑀)))) = 0 )

Theorem

cayleyhamiltonALT 22392*

Alternate proof of cayleyhamilton 22391, the Cayley-Hamilton theorem. This proof does not use cayleyhamilton0 22390 directly, but has the same structure as the proof of cayleyhamilton0 22390. In contrast to the proof of cayleyhamilton0 22390, 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.)

Theorem

cayleyhamilton1 22393*

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 22391, the meaning of "inserted" is made more transparent: If the characteristic polynomial is a polynomial with coefficients (𝐹‘𝑛), 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 = (0_g‘𝐴) & 𝐶 = (𝑁 CharPlyMat 𝑅) & 𝐾 = (coe₁‘(𝐶‘𝑀)) & ∗ = ( ·_𝑠 ‘𝐴) & ↑ = (._g‘(mulGrp‘𝐴)) & 𝐿 = (Base‘𝑅) & 𝑋 = (var₁‘𝑅) & 𝑃 = (Poly₁‘𝑅) & · = ( ·_𝑠 ‘𝑃) & 𝐸 = (._g‘(mulGrp‘𝑃)) & 𝑍 = (0_g‘𝑅) ⇒ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐹 ∈ (𝐿 ↑_m ℕ₀) ∧ 𝐹 finSupp 𝑍)) → ((𝐶‘𝑀) = (𝑃 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝐹‘𝑛) · (𝑛𝐸𝑋)))) → (𝐴 Σ_g (𝑛 ∈ ℕ₀ ↦ ((𝐹‘𝑛) ∗ (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 22415), 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 22395), then the function which associates with a set the set of topologies on it (df-topon 22412), and finally the class of topological spaces, as extensible structures having an underlying set and a topology on it (df-topsp 22434). Of course, a topology is the same thing as a topology on a set (see toprntopon 22426).

12.1.1.1 Topologies

Syntax

ctop 22394

Syntax for the class of topologies.

class Top

Definition

df-top 22395*

Define the class of topologies. It is a proper class (see topnex 22498). See istopg 22396 and istop2g 22397 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∪ 𝑦 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥)}

Theorem

istopg 22396*

Express the predicate "𝐽 is a topology". See istop2g 22397 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.)

(𝐽 ∈ 𝐴 → (𝐽 ∈ Top ↔ (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐽 (𝑥 ∩ 𝑦) ∈ 𝐽)))

Theorem

istop2g 22397*

Express the predicate "𝐽 is a topology" using nonempty finite intersections instead of binary intersections as in istopg 22396. (Contributed by NM, 19-Jul-2006.)

(𝐽 ∈ 𝐴 → (𝐽 ∈ Top ↔ (∀𝑥(𝑥 ⊆ 𝐽 → ∪ 𝑥 ∈ 𝐽) ∧ ∀𝑥((𝑥 ⊆ 𝐽 ∧ 𝑥 ≠ ∅ ∧ 𝑥 ∈ Fin) → ∩ 𝑥 ∈ 𝐽))))

Theorem

uniopn 22398

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 ∧ 𝐴 ⊆ 𝐽) → ∪ 𝐴 ∈ 𝐽)

Theorem

iunopn 22399*

The indexed union of a subset of a topology is an open set. (Contributed by NM, 5-Oct-2006.)

((𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽) → ∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐽)

Theorem

inopn 22400

The intersection of two open sets of a topology is an open set. (Contributed by NM, 17-Jul-2006.)

((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → (𝐴 ∩ 𝐵) ∈ 𝐽)

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