P. 229 of Theorem List - Metamath Proof Explorer

Type

Label

Description

Statement

Theorem

mp2pm2mplem3 22801*

Lemma 3 for mp2pm2mp 22804. (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 22802*

Lemma 4 for mp2pm2mp 22804. (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)    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐾 ∈ ℕ₀) → ((𝐼‘𝑂) decompPMat 𝐾) = ((coe₁‘𝑂)‘𝐾))

Theorem

mp2pm2mplem5 22803*

Lemma 5 for mp2pm2mp 22804. (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 22804*

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 22805*

Lemma 2 for pm2mpghm 22809. (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 22806

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 22807

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 22808

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 22809

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 22810

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 22811*

Lemma 1 for pm2mpmhm 22813. (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 22812*

Lemma 2 for pm2mpmhm 22813. (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 22813

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 22814

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 22815

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 22816

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 22817

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 22818*

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 22820) 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 22819

Extend class notation with the characteristic polynomial.

class CharPlyMat

Definition

df-chpmat 22820*

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 22821

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 22822

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 22823*

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 22824

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 22825

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 22826*

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 22827

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 22828

Lemma for chpmat1d 22829. (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 22829

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 22830

Lemma 0 for chpdmat 22834. (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 22831

Lemma 1 for chpdmat 22834. (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 22832

Lemma 2 for chpdmat 22834. (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 22833

Lemma 3 for chpdmat 22834. (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 22834*

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 22835*

The characteristic polynomial of a (nonempty!) scalar matrix. (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) = 𝐸)) → (𝐶‘𝑀) = ((♯‘𝑁) ↑ (𝑋 − (𝑆‘𝐸))))

Theorem

chpscmat0 22836*

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 22837*

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 22838*

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 22839

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 22840

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 22841

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 22871. 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 22859, cayhamlem3 22880 and cayhamlem4 22881.

Theorem

fvmptnn04if 22842*

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 22843*

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 22844*

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 22845*

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 22846*

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 22847*

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 = (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))))))))    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝐺:ℕ₀⟶(Base‘𝑌))

Theorem

chfacfisfcpmat 22848*

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 = (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))))))))    &   ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅)    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝐺:ℕ₀⟶𝑆)

Theorem

chfacffsupp 22849*

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 = (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))))))))    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → 𝐺 finSupp (0_g‘𝑌))

Theorem

chfacfscmulcl 22850*

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 22851*

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 22852*

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 22853*

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 22854*

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 = (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))    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝐾 ∈ ℕ₀) → ((𝐾 ↑ (𝑇‘𝑀)) × (𝐺‘𝐾)) ∈ (Base‘𝑌))

Theorem

chfacfpmmul0 22855*

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 = (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))    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠))) ∧ 𝐾 ∈ (ℤ_≥‘(𝑠 + 2))) → ((𝐾 ↑ (𝑇‘𝑀)) × (𝐺‘𝐾)) = 0 )

Theorem

chfacfpmmulfsupp 22856*

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 = (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))    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑖 ∈ ℕ₀ ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖))) finSupp 0 )

Theorem

chfacfpmmulgsum 22857*

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 22858*

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 22859*

Lemma 1 for cayleyhamilton 22883. (Contributed by AV, 11-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))    ⇒   ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑_m (0...𝑠)))) → (𝑌 Σ_g (𝑖 ∈ ℕ₀ ↦ ((𝑖 ↑ (𝑇‘𝑀)) × (𝐺‘𝑖)))) = 0 )

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 22820, 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 22824).

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 22824). 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 characteristic 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 22289), 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 22885. 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, XR	(var1‘𝑅)	The variable of polynomials over (the ring) 𝑅
𝑌, XA	(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)
×, xY	(.r‘𝑌)	The multiplication of polynomial matrices
xA	(.r‘𝐴)	The multiplication of matrices
xQ	(.r‘𝑄)	The multiplication of polynomials over matrices
1, 1Y	(1r‘𝑌)	The identity matrix in the algebra of polynomial matrices over 𝑅
1A	(1r‘𝐴)	The identity matrix in the algebra of matrices over 𝑅
0, 0Y	(0g‘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 22748 )
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 diagonal 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:

1. "c(M) e. B(Y)", or 𝐼 ∈ (Base‘𝑌), see chmatcl 22821
2. "p(M) e. B(P)", or 𝐾 ∈ (Base‘𝑃), see chpmatply1 22825
3. "T(M) e. B(Y)", or (𝑇‘𝑀) ∈ (Base‘𝑌), see mat2pmatbas 22719
4. 𝐽:(Base‘𝑌)⟶(Base‘𝑌), see maduf 22634
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 22781:
   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.
2. Write (𝐼 × 𝑊), the product of the characteristic matrix and its adjugate as finite sum of scaled monomials, see cpmadugsumfi 22870. 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 22781 (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 20287]
   = 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 22869]
   This step corresponds partially to (4) in Wikipedia.
3. Write (𝐼 × 𝑊) as infinite, but finitely supported sum of scaled monomials, see cpmadugsum 22871:
   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 22847, which covers the special terms and the padding with 0. G(i) is a constant polynomial matrix (see chfacfisfcpmat 22848). 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 diagonal elements, as infinite, but finitely supported sum of scaled monomials. See cpmidgsum 22861:
   p(m) *_Y I_Y = sum_i ( X_R ^i *_Y ( S(p_i) *_Y I_Y ) )
   The proof of cpmidgsum 22861 is making use of pmatcollpwscmat 22784, 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.
5. 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 22876]
   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 I_Y) = sum_i ( p_i *_A I_A ) *_Q X_A ^i ) [see cpmidpmat 22866]
7. Equate the sum representations resulting from steps 5. and 6. by using cpmadurid 22860 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 22860]
   = 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 22879, 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)) = p_i *_A I_A , see chcoeffeqlem 22878. 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 22879:
   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.
10. Transform the right hand side of the equation in step 9. into an appropriate form, see cayhamlem3 22880:
    sum_i ( p_i *_A M^i )
    = sum_i ( M^i x_A ( p_i *_A I_A) ) [see cayhamlem2 22877]
    = sum_i ( M^i x_A U(G(i)) ) [see chcoeffeq 22879]
11. Apply the theorem for telescoping sums, see telgsumfz 19988, 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 22859:
    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 22858]
    = ( 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 19988]
    = 0_Y [see grpnpncan0 19030] This step corresponds partially to (8) in Wikipedia.
12. Since 𝑇 is a ring homomorphism (see mat2pmatrhm 22727), 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 22881:
    sum_i ( p_i *_A M^i )
    = sum_i ( M^i x_A U(G(i)) ) [see cayhamlem3 22880 (step 10.)]
    = U(T₁(sum_i ( M^i x_A U(G(i)) ))) [see m2cpminvid 22746]
    = U(sum_i T₁( M^i x_A U(G(i)) )) [see gsummptmhm 19938]
    = U(sum_i ( T₁(M^i) x_Y T₁(U(G(i))) )) [see rhmmul 20468]
    = U(sum_i ( T₁(M)^i x_Y T₁(U(G(i))) )) [see mhmmulg 19109]
    = U(sum_i ( T₁(M)^i x_Y G(i) )) [see m2cpminvid2 22748 ]
    
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 22722 and mat2pmatrhm 22727) 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 22883 resp. cayleyhamilton0 22882:
    sum_i ( p_i *_A M^i )
    = U(sum_i ( T₁(M)^i x_Y G(i) )) [see cayhamlem4 22881 (step 12.)]
    = U(0_Y ) [see cayhamlem1 22859 (step 11.)]
    = 0_A [see m2cpminv0 22754]

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 22860

The right-hand fundamental relation of the adjugate (see madurid 22637) 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 22861*

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 22862*

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 22863*

Lemma 1 for cpmidpmat 22866. (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 22864*

Lemma 2 for cpmidpmat 22866. (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 22865*

Lemma 3 for cpmidpmat 22866. (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 22866*

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 22867*

Lemma B for cpmadugsum 22871. (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 22868*

Lemma C for cpmadugsum 22871. (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 22869*

Lemma F for cpmadugsum 22871. (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 22870*

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 22871*

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 22872*

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 22873*

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 22874*

Lemma 1 for cpmadumatpoly 22876. (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 22875*

Lemma 2 for cpmadumatpoly 22876. (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) finSupp (0_g‘𝐴))

Theorem

cpmadumatpoly 22876*

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 22877

Lemma for cayhamlem3 22880. (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 22878*

Lemma for chcoeffeq 22879. (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 22879*

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 = (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))∀𝑛 ∈ ℕ₀ (𝑈‘(𝐺‘𝑛)) = (((coe₁‘𝐾)‘𝑛) ∗ 1 ))

Theorem

cayhamlem3 22880*

Lemma for cayhamlem4 22881. (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 22881*

Lemma for cayleyhamilton 22883. (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 22882*

The Cayley-Hamilton theorem: A matrix over a commutative ring "satisfies its own characteristic equation". This version of cayleyhamilton 22883 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 22884)! (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 22883*

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 22884*

Alternate proof of cayleyhamilton 22883, the Cayley-Hamilton theorem. This proof does not use cayleyhamilton0 22882 directly, but has the same structure as the proof of cayleyhamilton0 22882. In contrast to the proof of cayleyhamilton0 22882, 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 = (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

cayleyhamilton1 22885*

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 22883, 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 22907), 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 22887), then the function which associates with a set the set of topologies on it (df-topon 22904), and finally the class of topological spaces, as extensible structures having an underlying set and a topology on it (df-topsp 22926). Of course, a topology is the same thing as a topology on a set (see toprntopon 22918).

12.1.1.1  Topologies

Syntax

ctop 22886

Syntax for the class of topologies.

class Top

Definition

df-top 22887*

Define the class of topologies. It is a proper class (see topnex 22990). See istopg 22888 and istop2g 22889 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 22888*

Express the predicate "𝐽 is a topology". See istop2g 22889 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 22889*

Express the predicate "𝐽 is a topology" using nonempty finite intersections instead of binary intersections as in istopg 22888. (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 22890

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 22891*

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 22892

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)

Theorem

fitop 22893

A topology is closed under finite intersections. (Contributed by Jeff Hankins, 7-Oct-2009.)

⊢ (𝐽 ∈ Top → (fi‘𝐽) = 𝐽)

Theorem

fiinopn 22894

The intersection of a nonempty finite family of open sets is open. (Contributed by FL, 20-Apr-2012.)

⊢ (𝐽 ∈ Top → ((𝐴 ⊆ 𝐽 ∧ 𝐴 ≠ ∅ ∧ 𝐴 ∈ Fin) → ∩ 𝐴 ∈ 𝐽))

Theorem

iinopn 22895*

The intersection of a nonempty finite family of open sets is open. (Contributed by Mario Carneiro, 14-Sep-2014.)

⊢ ((𝐽 ∈ Top ∧ (𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐽)) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐽)

Theorem

unopn 22896

The union of two open sets is open. (Contributed by Jeff Madsen, 2-Sep-2009.)

⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → (𝐴 ∪ 𝐵) ∈ 𝐽)

Theorem

0opn 22897

The empty set is an open subset of any topology. (Contributed by Stefan Allan, 27-Feb-2006.)

⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽)

Theorem

0ntop 22898

The empty set is not a topology. (Contributed by FL, 1-Jun-2008.)

⊢ ¬ ∅ ∈ Top

Theorem

topopn 22899

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

⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)

Theorem

eltopss 22900

A member of a topology is a subset of its underlying set. (Contributed by NM, 12-Sep-2006.)

⊢ 𝑋 = ∪ 𝐽    ⇒   ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ 𝑋)