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Theorem List for Metamath Proof Explorer - 20401-20500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempmatcollpw2lem 20401* Lemma for pmatcollpw2 20402. (Contributed by AV, 3-Oct-2019.) (Revised by AV, 3-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    × = ( ·𝑠𝑃)    &    = (.g‘(mulGrp‘𝑃))    &   𝑋 = (var1𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋)))) finSupp (0g𝐶))

Theorempmatcollpw2 20402* Write a polynomial matrix as sum of matrices whose entries are products of variable powers and constant polynomials collecting like powers. (Contributed by AV, 3-Oct-2019.) (Revised by AV, 3-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    × = ( ·𝑠𝑃)    &    = (.g‘(mulGrp‘𝑃))    &   𝑋 = (var1𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → 𝑀 = (𝐶 Σg (𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))))))

Theoremmonmatcollpw 20403 The matrix consisting of the coefficients in the polynomial entries of a polynomial matrix having scaled monomials with the same power as entries is the matrix of the coefficients of the monomials or a zero matrix. Generalization of decpmatid 20394 (but requires 𝑅 to be commutative!). (Contributed by AV, 11-Nov-2019.) (Revised by AV, 4-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐾 = (Base‘𝐴)    &    0 = (0g𝐴)    &    = (.g‘(mulGrp‘𝑃))    &   𝑋 = (var1𝑅)    &    · = ( ·𝑠𝐶)    &   𝑇 = (𝑁 matToPolyMat 𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0𝐼 ∈ ℕ0)) → (((𝐿 𝑋) · (𝑇𝑀)) decompPMat 𝐼) = if(𝐼 = 𝐿, 𝑀, 0 ))

Theorempmatcollpwlem 20404 Lemma for pmatcollpw 20405. (Contributed by AV, 26-Oct-2019.) (Revised by AV, 4-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝐶)    &    = (.g‘(mulGrp‘𝑃))    &   𝑋 = (var1𝑅)    &   𝑇 = (𝑁 matToPolyMat 𝑅)       ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎𝑁𝑏𝑁) → ((𝑎(𝑀 decompPMat 𝑛)𝑏)( ·𝑠𝑃)(𝑛 𝑋)) = (𝑎((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛)))𝑏))

Theorempmatcollpw 20405* Write a polynomial matrix (over a commutative ring) as sum of products of variable powers and constant matrices with scalar entries. (Contributed by AV, 26-Oct-2019.) (Revised by AV, 4-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝐶)    &    = (.g‘(mulGrp‘𝑃))    &   𝑋 = (var1𝑅)    &   𝑇 = (𝑁 matToPolyMat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑀 = (𝐶 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛))))))

Theorempmatcollpwfi 20406* Write a polynomial matrix (over a commutative ring) as a finite sum of products of variable powers and constant matrices with scalar entries. (Contributed by AV, 4-Nov-2019.) (Revised by AV, 4-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝐶)    &    = (.g‘(mulGrp‘𝑃))    &   𝑋 = (var1𝑅)    &   𝑇 = (𝑁 matToPolyMat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ∃𝑠 ∈ ℕ0 𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛))))))

Theorempmatcollpw3lem 20407* Lemma for pmatcollpw3 20408 and pmatcollpw3fi 20409: Write a polynomial matrix (over a commutative ring) as sum of products of variable powers and constant matrices with scalar entries. (Contributed by AV, 8-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝐶)    &    = (.g‘(mulGrp‘𝑃))    &   𝑋 = (var1𝑅)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐷 = (Base‘𝐴)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝐼 ⊆ ℕ0𝐼 ≠ ∅)) → (𝑀 = (𝐶 Σg (𝑛𝐼 ↦ ((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛))))) → ∃𝑓 ∈ (𝐷𝑚 𝐼)𝑀 = (𝐶 Σg (𝑛𝐼 ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛)))))))

Theorempmatcollpw3 20408* Write a polynomial matrix (over a commutative ring) as sum of products of variable powers and constant matrices with scalar entries. (Contributed by AV, 27-Oct-2019.) (Revised by AV, 4-Dec-2019.) (Proof shortened by AV, 8-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝐶)    &    = (.g‘(mulGrp‘𝑃))    &   𝑋 = (var1𝑅)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐷 = (Base‘𝐴)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ∃𝑓 ∈ (𝐷𝑚0)𝑀 = (𝐶 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛))))))

Theorempmatcollpw3fi 20409* Write a polynomial matrix (over a commutative ring) as a finite sum of products of variable powers and constant matrices with scalar entries. (Contributed by AV, 4-Nov-2019.) (Revised by AV, 4-Dec-2019.) (Proof shortened by AV, 8-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝐶)    &    = (.g‘(mulGrp‘𝑃))    &   𝑋 = (var1𝑅)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐷 = (Base‘𝐴)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ∃𝑠 ∈ ℕ0𝑓 ∈ (𝐷𝑚 (0...𝑠))𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛))))))

Theorempmatcollpw3fi1lem1 20410* Lemma 1 for pmatcollpw3fi1 20412. (Contributed by AV, 6-Nov-2019.) (Revised by AV, 4-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝐶)    &    = (.g‘(mulGrp‘𝑃))    &   𝑋 = (var1𝑅)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐷 = (Base‘𝐴)    &    0 = (0g𝐴)    &   𝐻 = (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝐺‘0), 0 ))       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷𝑚 {0}) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝐺𝑛)))))) → 𝑀 = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 𝑋) (𝑇‘(𝐻𝑛))))))

Theorempmatcollpw3fi1lem2 20411* Lemma 2 for pmatcollpw3fi1 20412. (Contributed by AV, 6-Nov-2019.) (Revised by AV, 4-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝐶)    &    = (.g‘(mulGrp‘𝑃))    &   𝑋 = (var1𝑅)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐷 = (Base‘𝐴)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (∃𝑓 ∈ (𝐷𝑚 {0})𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛))))) → ∃𝑠 ∈ ℕ ∃𝑓 ∈ (𝐷𝑚 (0...𝑠))𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛)))))))

Theorempmatcollpw3fi1 20412* Write a polynomial matrix (over a commutative ring) as a finite sum of (at least two) products of variable powers and constant matrices with scalar entries. (Contributed by AV, 6-Nov-2019.) (Revised by AV, 4-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝐶)    &    = (.g‘(mulGrp‘𝑃))    &   𝑋 = (var1𝑅)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐷 = (Base‘𝐴)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ∃𝑠 ∈ ℕ ∃𝑓 ∈ (𝐷𝑚 (0...𝑠))𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 𝑋) (𝑇‘(𝑓𝑛))))))

Theorempmatcollpwscmatlem1 20413 Lemma 1 for pmatcollpwscmat 20415. (Contributed by AV, 2-Nov-2019.) (Revised by AV, 4-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝐶)    &    = (.g‘(mulGrp‘𝑃))    &   𝑋 = (var1𝑅)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐷 = (Base‘𝐴)    &   𝑈 = (algSc‘𝑃)    &   𝐾 = (Base‘𝑅)    &   𝐸 = (Base‘𝑃)    &   𝑆 = (algSc‘𝑃)    &    1 = (1r𝐶)    &   𝑀 = (𝑄 1 )       ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) ∧ (𝑎𝑁𝑏𝑁)) → (((coe1‘(𝑎𝑀𝑏))‘𝐿)( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))) = if(𝑎 = 𝑏, (𝑈‘((coe1𝑄)‘𝐿)), (0g𝑃)))

Theorempmatcollpwscmatlem2 20414 Lemma 2 for pmatcollpwscmat 20415. (Contributed by AV, 2-Nov-2019.) (Revised by AV, 4-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝐶)    &    = (.g‘(mulGrp‘𝑃))    &   𝑋 = (var1𝑅)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐷 = (Base‘𝐴)    &   𝑈 = (algSc‘𝑃)    &   𝐾 = (Base‘𝑅)    &   𝐸 = (Base‘𝑃)    &   𝑆 = (algSc‘𝑃)    &    1 = (1r𝐶)    &   𝑀 = (𝑄 1 )       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0𝑄𝐸)) → (𝑇‘(𝑀 decompPMat 𝐿)) = ((𝑈‘((coe1𝑄)‘𝐿)) 1 ))

Theorempmatcollpwscmat 20415* Write a scalar matrix over polynomials (over a commutative ring) as sum of the product of variable powers and constant scalar matrices with scalar entries. (Contributed by AV, 2-Nov-2019.) (Revised by AV, 4-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝐶)    &    = (.g‘(mulGrp‘𝑃))    &   𝑋 = (var1𝑅)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐷 = (Base‘𝐴)    &   𝑈 = (algSc‘𝑃)    &   𝐾 = (Base‘𝑅)    &   𝐸 = (Base‘𝑃)    &   𝑆 = (algSc‘𝑃)    &    1 = (1r𝐶)    &   𝑀 = (𝑄 1 )       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑄𝐸) → 𝑀 = (𝐶 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 𝑋) ((𝑈‘((coe1𝑄)‘𝑛)) 1 )))))

11.4.4  Ring isomorphism between polynomial matrices and polynomials over matrices

The main result of this section is theorem pmmpric 20447, which shows that the ring of polynomial matrices and the ring of polynomials having matrices as coefficients (called "polynomials over matrices" in the following) are isomorphic:
(Poly1‘(𝑁 Mat 𝑅)) ≃ (𝑁 Mat (Poly1𝑅))

Or in a more common notation:
(𝑁 Mat (Poly1𝑅)) corresponds to M(n, R[t]), the ring of n x n polynomial matrices over the ring R.
(Poly1‘(𝑁 Mat 𝑅)) corresponds to M(n, R)[t], the polynomial ring over the ring of n x n matrices with entries in ring R.

𝑇 = (𝑚𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) (𝑘 𝑋)))))

with 𝐵 = (Base‘(𝑁 Mat (Poly1𝑅))) and (𝑚 decompPMat 𝑘) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1 ( i m j ) ) 𝑘))) is an isomorphism between these rings:

𝑇:𝐵1-1-onto𝐿 with 𝐿 = (Base‘(Poly1‘(𝑁 Mat 𝑅))) (see pm2mpf1o 20439 and pm2mprngiso 20446), and

𝐼 = (𝑝𝐿 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1 p ) 𝑘)𝑗) · (𝑘𝐸𝑌))))))

is the corresponding inverse function:

(𝑇‘(𝐼𝑂)) = 𝑂) (see mp2pm2mp 20435).

In this section, the following conventions are mostly used:

• 𝑅 is a (unital) ring (see df-ring 18372)
• 𝑃 = (Poly1𝑅) is the polynomial algebra over (the ring) 𝑅 (see df-ply1 19373)
• 𝐾 = (Base‘𝑃) is its base set (see df-base 15700)
• 𝑌 = (var1𝑅) is its variable (see df-vr1 19372)
• · = ( ·𝑠𝑃) is its scalar multiplication (see df-vsca 15785 or lmodvscl 18703)
• 𝐸 = (.g‘(mulGrp‘𝑃)) is its exponentiation (see df-mulg 17364)
• 𝐴 = (𝑁 Mat 𝑅) is the algebra of N x N matrices over (the ring) 𝑅 (see df-mat 20033)
• 𝐶 = (𝑁 Mat 𝑃) is the algebra of N x N matrices over (the polynomial ring) 𝑃.
• 𝐵 = (Base‘𝐶) is its base set
• 𝑀𝐵 is a concrete polynomial matrix
• 𝑄 = (Poly1𝐴) is the polynomial algebra over (the matrix ring) 𝐴.
• 𝐿 = (Base‘𝑄) is its base set
• 𝑂𝐿 is a concrete polynomial with matrix coefficients
• 𝑋 = (var1𝐴) is its variable
• = ( ·𝑠𝑄) is its scalar multiplication
• = (.g‘(mulGrp‘𝑄)) is its exponentiation

Syntaxcpm2mp 20416 Extend class notation with the transformation of a polynomial matrix into a polynomial over matrices.
class pMatToMatPoly

Definitiondf-pm2mp 20417* Transformation of a polynomial matrix (over a ring) into a polynomial over matrices (over the same ring). (Contributed by AV, 5-Dec-2019.)
pMatToMatPoly = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat (Poly1𝑟))) ↦ (𝑛 Mat 𝑟) / 𝑎(Poly1𝑎) / 𝑞(𝑞 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠𝑞)(𝑘(.g‘(mulGrp‘𝑞))(var1𝑎)))))))

Theorempm2mpf1lem 20418* Lemma for pm2mpf1 20423. (Contributed by AV, 14-Oct-2019.) (Revised by AV, 4-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑋 = (var1𝐴)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑈𝐵𝐾 ∈ ℕ0)) → ((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑈 decompPMat 𝑘) (𝑘 𝑋)))))‘𝐾) = (𝑈 decompPMat 𝐾))

Theorempm2mpval 20419* Value of the transformation of a polynomial matrix into a polynomial over matrices. (Contributed by AV, 5-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑋 = (var1𝐴)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝑇 = (𝑁 pMatToMatPoly 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑇 = (𝑚𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) (𝑘 𝑋))))))

Theorempm2mpfval 20420* A polynomial matrix transformed into a polynomial over matrices. (Contributed by AV, 4-Oct-2019.) (Revised by AV, 5-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑋 = (var1𝐴)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝑇 = (𝑁 pMatToMatPoly 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → (𝑇𝑀) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) (𝑘 𝑋)))))

Theorempm2mpcl 20421 The transformation of polynomial matrices into polynomials over matrices maps polynomial matrices to polynomials over matrices. (Contributed by AV, 5-Oct-2019.) (Revised by AV, 5-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑋 = (var1𝐴)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝑇 = (𝑁 pMatToMatPoly 𝑅)    &   𝐿 = (Base‘𝑄)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑇𝑀) ∈ 𝐿)

Theorempm2mpf 20422 The transformation of polynomial matrices into polynomials over matrices is a function mapping polynomial matrices to polynomials over matrices. (Contributed by AV, 5-Oct-2019.) (Revised by AV, 5-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑋 = (var1𝐴)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝑇 = (𝑁 pMatToMatPoly 𝑅)    &   𝐿 = (Base‘𝑄)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵𝐿)

Theorempm2mpf1 20423 The transformation of polynomial matrices into polynomials over matrices is a 1-1 function mapping polynomial matrices to polynomials over matrices. (Contributed by AV, 14-Oct-2019.) (Revised by AV, 6-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑋 = (var1𝐴)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝑇 = (𝑁 pMatToMatPoly 𝑅)    &   𝐿 = (Base‘𝑄)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵1-1𝐿)

Theorempm2mpcoe1 20424 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑋 = (var1𝐴)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝑇 = (𝑁 pMatToMatPoly 𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀𝐵𝐾 ∈ ℕ0)) → ((coe1‘(𝑇𝑀))‘𝐾) = (𝑀 decompPMat 𝐾))

Theoremidpm2idmp 20425 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑋 = (var1𝐴)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝑇 = (𝑁 pMatToMatPoly 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑇‘(1r𝐶)) = (1r𝑄))

Theoremmptcoe1matfsupp 20426* The mapping extracting the entries of the coefficient matrices of a polynomial over matrices at a fixed position is finitely supported. (Contributed by AV, 6-Oct-2019.) (Proof shortened by AV, 23-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝐼𝑁𝐽𝑁) → (𝑘 ∈ ℕ0 ↦ (𝐼((coe1𝑂)‘𝑘)𝐽)) finSupp (0g𝑅))

Theoremmply1topmatcllem 20427* Lemma for mply1topmatcl 20429. (Contributed by AV, 6-Oct-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)    &   𝑃 = (Poly1𝑅)    &    · = ( ·𝑠𝑃)    &   𝐸 = (.g‘(mulGrp‘𝑃))    &   𝑌 = (var1𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝐼𝑁𝐽𝑁) → (𝑘 ∈ ℕ0 ↦ ((𝐼((coe1𝑂)‘𝑘)𝐽) · (𝑘𝐸𝑌))) finSupp (0g𝑃))

Theoremmply1topmatval 20428* A polynomial over matrices transformed into a polynomial matrix. 𝐼 is the inverse function of the transformation 𝑇 of polynomial matrices into polynomials over matrices: (𝑇‘(𝐼𝑂)) = 𝑂) (see mp2pm2mp 20435). (Contributed by AV, 6-Oct-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)    &   𝑃 = (Poly1𝑅)    &    · = ( ·𝑠𝑃)    &   𝐸 = (.g‘(mulGrp‘𝑃))    &   𝑌 = (var1𝑅)    &   𝐼 = (𝑝𝐿 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))       ((𝑁𝑉𝑂𝐿) → (𝐼𝑂) = (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))

Theoremmply1topmatcl 20429* A polynomial over matrices transformed into a polynomial matrix is a polynomial matrix. (Contributed by AV, 6-Oct-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)    &   𝑃 = (Poly1𝑅)    &    · = ( ·𝑠𝑃)    &   𝐸 = (.g‘(mulGrp‘𝑃))    &   𝑌 = (var1𝑅)    &   𝐼 = (𝑝𝐿 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝐼𝑂) ∈ 𝐵)

Theoremmp2pm2mplem1 20430* Lemma 1 for mp2pm2mp 20435. (Contributed by AV, 9-Oct-2019.) (Revised by AV, 5-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)    &    · = ( ·𝑠𝑃)    &   𝐸 = (.g‘(mulGrp‘𝑃))    &   𝑌 = (var1𝑅)    &   𝐼 = (𝑝𝐿 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝐼𝑂) = (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))

Theoremmp2pm2mplem2 20431* Lemma 2 for mp2pm2mp 20435. (Contributed by AV, 10-Oct-2019.) (Revised by AV, 5-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)    &    · = ( ·𝑠𝑃)    &   𝐸 = (.g‘(mulGrp‘𝑃))    &   𝑌 = (var1𝑅)    &   𝐼 = (𝑝𝐿 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))    &   𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))))) ∈ 𝐵)

Theoremmp2pm2mplem3 20432* Lemma 3 for mp2pm2mp 20435. (Contributed by AV, 10-Oct-2019.) (Revised by AV, 5-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)    &    · = ( ·𝑠𝑃)    &   𝐸 = (.g‘(mulGrp‘𝑃))    &   𝑌 = (var1𝑅)    &   𝐼 = (𝑝𝐿 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))    &   𝑃 = (Poly1𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝐾 ∈ ℕ0) → ((𝐼𝑂) decompPMat 𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌)))))‘𝐾)))

Theoremmp2pm2mplem4 20433* Lemma 4 for mp2pm2mp 20435. (Contributed by AV, 12-Oct-2019.) (Revised by AV, 5-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)    &    · = ( ·𝑠𝑃)    &   𝐸 = (.g‘(mulGrp‘𝑃))    &   𝑌 = (var1𝑅)    &   𝐼 = (𝑝𝐿 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))    &   𝑃 = (Poly1𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) ∧ 𝐾 ∈ ℕ0) → ((𝐼𝑂) decompPMat 𝐾) = ((coe1𝑂)‘𝐾))

Theoremmp2pm2mplem5 20434* Lemma 5 for mp2pm2mp 20435. (Contributed by AV, 12-Oct-2019.) (Revised by AV, 5-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)    &    · = ( ·𝑠𝑃)    &   𝐸 = (.g‘(mulGrp‘𝑃))    &   𝑌 = (var1𝑅)    &   𝐼 = (𝑝𝐿 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))    &   𝑃 = (Poly1𝑅)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑋 = (var1𝐴)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝑘 ∈ ℕ0 ↦ (((𝐼𝑂) decompPMat 𝑘) (𝑘 𝑋))) finSupp (0g𝑄))

Theoremmp2pm2mp 20435* A polynomial over matrices transformed into a polynomial matrix transformed back into the polynomial over matrices. (Contributed by AV, 12-Oct-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)    &    · = ( ·𝑠𝑃)    &   𝐸 = (.g‘(mulGrp‘𝑃))    &   𝑌 = (var1𝑅)    &   𝐼 = (𝑝𝐿 ↦ (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌))))))    &   𝑃 = (Poly1𝑅)    &   𝑇 = (𝑁 pMatToMatPoly 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂𝐿) → (𝑇‘(𝐼𝑂)) = 𝑂)

Theorempm2mpghmlem2 20436* Lemma 2 for pm2mpghm 20440. (Contributed by AV, 15-Oct-2019.) (Revised by AV, 4-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑋 = (var1𝐴)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) (𝑘 𝑋))) finSupp (0g𝑄))

Theorempm2mpghmlem1 20437 Lemma 1 for pm2mpghm . (Contributed by AV, 15-Oct-2019.) (Revised by AV, 4-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑋 = (var1𝐴)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝐾 ∈ ℕ0) → ((𝑀 decompPMat 𝐾) (𝐾 𝑋)) ∈ 𝐿)

Theorempm2mpfo 20438 The transformation of polynomial matrices into polynomials over matrices is a function mapping polynomial matrices onto polynomials over matrices. (Contributed by AV, 12-Oct-2019.) (Revised by AV, 6-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑋 = (var1𝐴)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)    &   𝑇 = (𝑁 pMatToMatPoly 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵onto𝐿)

Theorempm2mpf1o 20439 The transformation of polynomial matrices into polynomials over matrices is a 1-1 function mapping polynomial matrices onto polynomials over matrices. (Contributed by AV, 14-Oct-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑋 = (var1𝐴)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)    &   𝑇 = (𝑁 pMatToMatPoly 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵1-1-onto𝐿)

Theorempm2mpghm 20440 The transformation of polynomial matrices into polynomials over matrices is an additive group homomorphism. (Contributed by AV, 16-Oct-2019.) (Revised by AV, 6-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑋 = (var1𝐴)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)    &   𝑇 = (𝑁 pMatToMatPoly 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐶 GrpHom 𝑄))

Theorempm2mpgrpiso 20441 The transformation of polynomial matrices into polynomials over matrices is an additive group isomorphism. (Contributed by AV, 17-Oct-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑋 = (var1𝐴)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)    &   𝑇 = (𝑁 pMatToMatPoly 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐶 GrpIso 𝑄))

Theorempm2mpmhmlem1 20442* Lemma 1 for pm2mpmhm 20444. (Contributed by AV, 21-Oct-2019.) (Revised by AV, 6-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑋 = (var1𝐴)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝐿 = (Base‘𝑄)    &   𝑇 = (𝑁 pMatToMatPoly 𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑙 ∈ ℕ0 ↦ ((𝐴 Σg (𝑘 ∈ (0...𝑙) ↦ ((𝑥 decompPMat 𝑘)(.r𝐴)(𝑦 decompPMat (𝑙𝑘))))) (𝑙 𝑋))) finSupp (0g𝑄))

Theorempm2mpmhmlem2 20443* Lemma 2 for pm2mpmhm 20444. (Contributed by AV, 22-Oct-2019.) (Revised by AV, 6-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝑇 = (𝑁 pMatToMatPoly 𝑅)    &   𝐵 = (Base‘𝐶)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥𝐵𝑦𝐵 (𝑇‘(𝑥(.r𝐶)𝑦)) = ((𝑇𝑥)(.r𝑄)(𝑇𝑦)))

Theorempm2mpmhm 20444 The transformation of polynomial matrices into polynomials over matrices is a homomorphism of multiplicative monoids. (Contributed by AV, 22-Oct-2019.) (Revised by AV, 6-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝑇 = (𝑁 pMatToMatPoly 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ ((mulGrp‘𝐶) MndHom (mulGrp‘𝑄)))

Theorempm2mprhm 20445 The transformation of polynomial matrices into polynomials over matrices is a ring homomorphism. (Contributed by AV, 22-Oct-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝑇 = (𝑁 pMatToMatPoly 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐶 RingHom 𝑄))

Theorempm2mprngiso 20446 The transformation of polynomial matrices into polynomials over matrices is a ring isomorphism. (Contributed by AV, 22-Oct-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)    &   𝑇 = (𝑁 pMatToMatPoly 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐶 RingIso 𝑄))

Theorempmmpric 20447 The ring of polynomial matrices over a ring is isomorphic to the ring of polynomials over matrices of the same dimension over the same ring. (Contributed by AV, 30-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (Poly1𝐴)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶𝑟 𝑄)

Theoremmonmat2matmon 20448 The transformation of a polynomial matrix having scaled monomials with the same power as entries into a scaled monomial as a polynomial over matrices. (Contributed by AV, 11-Nov-2019.) (Revised by AV, 7-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑋 = (var1𝐴)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐾 = (Base‘𝐴)    &   𝑄 = (Poly1𝐴)    &   𝐼 = (𝑁 pMatToMatPoly 𝑅)    &   𝐸 = (.g‘(mulGrp‘𝑃))    &   𝑌 = (var1𝑅)    &    · = ( ·𝑠𝐶)    &   𝑇 = (𝑁 matToPolyMat 𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐾𝐿 ∈ ℕ0)) → (𝐼‘((𝐿𝐸𝑌) · (𝑇𝑀))) = (𝑀 (𝐿 𝑋)))

Theorempm2mp 20449* The transformation of a sum of matrices having scaled monomials with the same power as entries into a sum of scaled monomials as a polynomial over matrices. (Contributed by AV, 12-Nov-2019.) (Revised by AV, 7-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝑄)    &    = (.g‘(mulGrp‘𝑄))    &   𝑋 = (var1𝐴)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐾 = (Base‘𝐴)    &   𝑄 = (Poly1𝐴)    &   𝐼 = (𝑁 pMatToMatPoly 𝑅)    &   𝐸 = (.g‘(mulGrp‘𝑃))    &   𝑌 = (var1𝑅)    &    · = ( ·𝑠𝐶)    &   𝑇 = (𝑁 matToPolyMat 𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾𝑚0) ∧ 𝑀 finSupp (0g𝐴))) → (𝐼‘(𝐶 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸𝑌) · (𝑇‘(𝑀𝑛)))))) = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ ((𝑀𝑛) (𝑛 𝑋)))))

11.5  The characteristic polynomial

According to Wikipedia ("Characteristic polynomial", 31-Jul-2019, https://en.wikipedia.org/wiki/Characteristic_polynomial): "In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix as coefficients.". Based on the definition of the characteristic polynomial of a square matrix (df-chpmat 20451) the eigenvalues and corresponding eigenvectors can be defined.

11.5.1  Definition and basic properties

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

Syntaxcchpmat 20450 Extend class notation with the characteristic polynomial.
class CharPlyMat

Definitiondf-chpmat 20451* Define the characteristic polynomial of a square matrix. According to Wikipedia ("Characteristic polynomial", 31-Jul-2019, https://en.wikipedia.org/wiki/Characteristic_polynomial): "The characteristic polynomial of [an n x n matrix] A, denoted by pA(t), is the polynomial defined by pA ( t ) = det ( t I - A ) where I denotes the n-by-n identity matrix.". In addition, however, the underlying ring must be commutative, see definition in [Lang], p. 561: " Let k be a commutative ring ... Let M be any n x n matrix in k ... We define the characteristic polynomial PM(t) to be the determinant det ( t In - M ) where In is the unit n x n matrix." To be more precise, the matrices A and I on the right hand side are matrices with coefficients of a polynomial ring. Therefore, the original matrix A over a given commutative ring must be transformed into corresponding matrices over the polynomial ring over the given ring. (Contributed by AV, 2-Aug-2019.)
CharPlyMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ ((𝑛 maDet (Poly1𝑟))‘(((var1𝑟)( ·𝑠 ‘(𝑛 Mat (Poly1𝑟)))(1r‘(𝑛 Mat (Poly1𝑟))))(-g‘(𝑛 Mat (Poly1𝑟)))((𝑛 matToPolyMat 𝑟)‘𝑚)))))

Theoremchmatcl 20452 Closure of the characteristic matrix of a matrix. (Contributed by AV, 25-Oct-2019.) (Proof shortened by AV, 29-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝑋 = (var1𝑅)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &    = (-g𝑌)    &    · = ( ·𝑠𝑌)    &    1 = (1r𝑌)    &   𝐻 = ((𝑋 · 1 ) (𝑇𝑀))       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → 𝐻 ∈ (Base‘𝑌))

Theoremchmatval 20453 The entries of the characteristic matrix of a matrix. (Contributed by AV, 2-Aug-2019.) (Proof shortened by AV, 10-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝑋 = (var1𝑅)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &    = (-g𝑌)    &    · = ( ·𝑠𝑌)    &    1 = (1r𝑌)    &   𝐻 = ((𝑋 · 1 ) (𝑇𝑀))    &    = (-g𝑃)    &    0 = (0g𝑃)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝐼𝑁𝐽𝑁)) → (𝐼𝐻𝐽) = if(𝐼 = 𝐽, (𝑋 (𝐼(𝑇𝑀)𝐽)), ( 0 (𝐼(𝑇𝑀)𝐽))))

Theoremchpmatfval 20454* Value of the characteristic polynomial function. (Contributed by AV, 2-Aug-2019.)
𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝐷 = (𝑁 maDet 𝑃)    &    = (-g𝑌)    &   𝑋 = (var1𝑅)    &    · = ( ·𝑠𝑌)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &    1 = (1r𝑌)       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝐶 = (𝑚𝐵 ↦ (𝐷‘((𝑋 · 1 ) (𝑇𝑚)))))

Theoremchpmatval 20455 The characteristic polynomial of a (square) matrix (expressed with a determinant). (Contributed by AV, 2-Aug-2019.)
𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝐷 = (𝑁 maDet 𝑃)    &    = (-g𝑌)    &   𝑋 = (var1𝑅)    &    · = ( ·𝑠𝑌)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &    1 = (1r𝑌)       ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → (𝐶𝑀) = (𝐷‘((𝑋 · 1 ) (𝑇𝑀))))

Theoremchpmatply1 20456 The characteristic polynomial of a (square) matrix over a commutative ring is a polynomial, see also the following remark in [Lang], p. 561: "[the characteristic polynomial] is an element of k[t]". (Contributed by AV, 2-Aug-2019.) (Proof shortened by AV, 29-Nov-2019.)
𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝐸 = (Base‘𝑃)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝐶𝑀) ∈ 𝐸)

Theoremchpmatval2 20457* The characteristic polynomial of a (square) matrix (expressed with the Leibnitz formula for the determinant). (Contributed by AV, 2-Aug-2019.)
𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &    = (-g𝑌)    &   𝑋 = (var1𝑅)    &    · = ( ·𝑠𝑌)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &    1 = (1r𝑌)    &   𝐺 = (SymGrp‘𝑁)    &   𝐻 = (Base‘𝐺)    &   𝑍 = (ℤRHom‘𝑃)    &   𝑆 = (pmSgn‘𝑁)    &   𝑈 = (mulGrp‘𝑃)    &    × = (.r𝑃)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝐶𝑀) = (𝑃 Σg (𝑝𝐻 ↦ (((𝑍𝑆)‘𝑝) × (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)((𝑋 · 1 ) (𝑇𝑀))𝑥)))))))

Theoremchpmat0d 20458 The characteristic polynomial of the empty matrix. (Contributed by AV, 6-Aug-2019.)
𝐶 = (∅ CharPlyMat 𝑅)       (𝑅 ∈ Ring → (𝐶‘∅) = (1r‘(Poly1𝑅)))

Theoremchpmat1dlem 20459 Lemma for chpmat1d 20460. (Contributed by AV, 7-Aug-2019.)
𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑋 = (var1𝑅)    &    = (-g𝑃)    &   𝑆 = (algSc‘𝑃)    &   𝐺 = (𝑁 Mat 𝑃)    &   𝑇 = (𝑁 matToPolyMat 𝑅)       ((𝑅 ∈ Ring ∧ (𝑁 = {𝐼} ∧ 𝐼𝑉) ∧ 𝑀𝐵) → (𝐼((𝑋( ·𝑠𝐺)(1r𝐺))(-g𝐺)(𝑇𝑀))𝐼) = (𝑋 (𝑆‘(𝐼𝑀𝐼))))

Theoremchpmat1d 20460 The characteristic polynomial of a matrix with dimension 1. (Contributed by AV, 7-Aug-2019.)
𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑋 = (var1𝑅)    &    = (-g𝑃)    &   𝑆 = (algSc‘𝑃)       ((𝑅 ∈ CRing ∧ (𝑁 = {𝐼} ∧ 𝐼𝑉) ∧ 𝑀𝐵) → (𝐶𝑀) = (𝑋 (𝑆‘(𝐼𝑀𝐼))))

Theoremchpdmatlem0 20461 Lemma 0 for chpdmat 20465. (Contributed by AV, 18-Aug-2019.)
𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑆 = (algSc‘𝑃)    &   𝐵 = (Base‘𝐴)    &   𝑋 = (var1𝑅)    &    0 = (0g𝑅)    &   𝐺 = (mulGrp‘𝑃)    &    = (-g𝑃)    &   𝑄 = (𝑁 Mat 𝑃)    &    1 = (1r𝑄)    &    · = ( ·𝑠𝑄)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑋 · 1 ) ∈ (Base‘𝑄))

Theoremchpdmatlem1 20462 Lemma 1 for chpdmat 20465. (Contributed by AV, 18-Aug-2019.)
𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑆 = (algSc‘𝑃)    &   𝐵 = (Base‘𝐴)    &   𝑋 = (var1𝑅)    &    0 = (0g𝑅)    &   𝐺 = (mulGrp‘𝑃)    &    = (-g𝑃)    &   𝑄 = (𝑁 Mat 𝑃)    &    1 = (1r𝑄)    &    · = ( ·𝑠𝑄)    &   𝑍 = (-g𝑄)    &   𝑇 = (𝑁 matToPolyMat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → ((𝑋 · 1 )𝑍(𝑇𝑀)) ∈ (Base‘𝑄))

Theoremchpdmatlem2 20463 Lemma 2 for chpdmat 20465. (Contributed by AV, 18-Aug-2019.)
𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑆 = (algSc‘𝑃)    &   𝐵 = (Base‘𝐴)    &   𝑋 = (var1𝑅)    &    0 = (0g𝑅)    &   𝐺 = (mulGrp‘𝑃)    &    = (-g𝑃)    &   𝑄 = (𝑁 Mat 𝑃)    &    1 = (1r𝑄)    &    · = ( ·𝑠𝑄)    &   𝑍 = (-g𝑄)    &   𝑇 = (𝑁 matToPolyMat 𝑅)       ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑖𝑁) ∧ 𝑗𝑁) ∧ 𝑖𝑗) ∧ (𝑖𝑀𝑗) = 0 ) → (𝑖((𝑋 · 1 )𝑍(𝑇𝑀))𝑗) = (0g𝑃))

Theoremchpdmatlem3 20464 Lemma 3 for chpdmat 20465. (Contributed by AV, 18-Aug-2019.)
𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑆 = (algSc‘𝑃)    &   𝐵 = (Base‘𝐴)    &   𝑋 = (var1𝑅)    &    0 = (0g𝑅)    &   𝐺 = (mulGrp‘𝑃)    &    = (-g𝑃)    &   𝑄 = (𝑁 Mat 𝑃)    &    1 = (1r𝑄)    &    · = ( ·𝑠𝑄)    &   𝑍 = (-g𝑄)    &   𝑇 = (𝑁 matToPolyMat 𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝐾𝑁) → (𝐾((𝑋 · 1 )𝑍(𝑇𝑀))𝐾) = (𝑋 (𝑆‘(𝐾𝑀𝐾))))

Theoremchpdmat 20465* The characteristic polynomial of a diagonal matrix. (Contributed by AV, 18-Aug-2019.) (Proof shortened by AV, 21-Nov-2019.)
𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑆 = (algSc‘𝑃)    &   𝐵 = (Base‘𝐴)    &   𝑋 = (var1𝑅)    &    0 = (0g𝑅)    &   𝐺 = (mulGrp‘𝑃)    &    = (-g𝑃)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑀𝑗) = 0 )) → (𝐶𝑀) = (𝐺 Σg (𝑘𝑁 ↦ (𝑋 (𝑆‘(𝑘𝑀𝑘))))))

Theoremchpscmat 20466* The characteristic polynomial of a (nonempty!) scalar matrix. (Contributed by AV, 21-Aug-2019.)
𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑋 = (var1𝑅)    &   𝐺 = (mulGrp‘𝑃)    &    = (.g𝐺)    &   𝐷 = {𝑚 ∈ (Base‘𝐴) ∣ ∃𝑐 ∈ (Base‘𝑅)∀𝑖𝑁𝑗𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g𝑅))}    &   𝑆 = (algSc‘𝑃)    &    = (-g𝑃)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐷𝐼𝑁 ∧ ∀𝑛𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝐶𝑀) = ((#‘𝑁) (𝑋 (𝑆𝐸))))

Theoremchpscmat0 20467* The characteristic polynomial of a (nonempty!) scalar matrix, expressed with its diagonal element. (Contributed by AV, 21-Aug-2019.)
𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑋 = (var1𝑅)    &   𝐺 = (mulGrp‘𝑃)    &    = (.g𝐺)    &   𝐷 = {𝑚 ∈ (Base‘𝐴) ∣ ∃𝑐 ∈ (Base‘𝑅)∀𝑖𝑁𝑗𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g𝑅))}    &   𝑆 = (algSc‘𝑃)    &    = (-g𝑃)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐷𝐼𝑁 ∧ ∀𝑛𝑁 (𝑛𝑀𝑛) = (𝐼𝑀𝐼))) → (𝐶𝑀) = ((#‘𝑁) (𝑋 (𝑆‘(𝐼𝑀𝐼)))))

Theoremchpscmatgsumbin 20468* The characteristic polynomial of a (nonempty!) scalar matrix, expressed as finite group sum of binomials. (Contributed by AV, 2-Sep-2019.)
𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑋 = (var1𝑅)    &   𝐺 = (mulGrp‘𝑃)    &    = (.g𝐺)    &   𝐷 = {𝑚 ∈ (Base‘𝐴) ∣ ∃𝑐 ∈ (Base‘𝑅)∀𝑖𝑁𝑗𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g𝑅))}    &   𝑆 = (algSc‘𝑃)    &    = (-g𝑃)    &   𝐹 = (.g𝑃)    &   𝐻 = (mulGrp‘𝑅)    &   𝐸 = (.g𝐻)    &   𝐼 = (invg𝑅)    &    · = ( ·𝑠𝑃)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐷𝐽𝑁 ∧ ∀𝑛𝑁 (𝑛𝑀𝑛) = (𝐽𝑀𝐽))) → (𝐶𝑀) = (𝑃 Σg (𝑙 ∈ (0...(#‘𝑁)) ↦ (((#‘𝑁)C𝑙)𝐹((((#‘𝑁) − 𝑙)𝐸(𝐼‘(𝐽𝑀𝐽))) · (𝑙 𝑋))))))

Theoremchpscmatgsummon 20469* The characteristic polynomial of a (nonempty!) scalar matrix, expressed as finite group sum of scaled monomials. (Contributed by AV, 2-Sep-2019.)
𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑋 = (var1𝑅)    &   𝐺 = (mulGrp‘𝑃)    &    = (.g𝐺)    &   𝐷 = {𝑚 ∈ (Base‘𝐴) ∣ ∃𝑐 ∈ (Base‘𝑅)∀𝑖𝑁𝑗𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g𝑅))}    &   𝑆 = (algSc‘𝑃)    &    = (-g𝑃)    &   𝐹 = (.g𝑃)    &   𝐻 = (mulGrp‘𝑅)    &   𝐸 = (.g𝐻)    &   𝐼 = (invg𝑅)    &    · = ( ·𝑠𝑃)    &   𝑍 = (.g𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀𝐷𝐽𝑁 ∧ ∀𝑛𝑁 (𝑛𝑀𝑛) = (𝐽𝑀𝐽))) → (𝐶𝑀) = (𝑃 Σg (𝑙 ∈ (0...(#‘𝑁)) ↦ ((((#‘𝑁)C𝑙)𝑍(((#‘𝑁) − 𝑙)𝐸(𝐼‘(𝐽𝑀𝐽)))) · (𝑙 𝑋)))))

Theoremchp0mat 20470 The characteristic polynomial of the zero matrix. (Contributed by AV, 18-Aug-2019.)
𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑋 = (var1𝑅)    &   𝐺 = (mulGrp‘𝑃)    &    = (.g𝐺)    &    0 = (0g𝐴)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝐶0 ) = ((#‘𝑁) 𝑋))

Theoremchpidmat 20471 The characteristic polynomial of the identity matrix. (Contributed by AV, 19-Aug-2019.)
𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑋 = (var1𝑅)    &   𝐺 = (mulGrp‘𝑃)    &    = (.g𝐺)    &   𝐼 = (1r𝐴)    &   𝑆 = (algSc‘𝑃)    &    1 = (1r𝑅)    &    = (-g𝑃)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝐶𝐼) = ((#‘𝑁) (𝑋 (𝑆1 ))))

Theoremchmaidscmat 20472 The characteristic polynomial of a matrix multiplied with the identity matrix is a scalar matrix. (Contributed by AV, 30-Oct-2019.) (Revised by AV, 19-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐸 = (Base‘𝑃)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝐾 = (Base‘𝑌)    &    · = ( ·𝑠𝑌)    &    0 = (0g𝑃)    &    1 = (1r𝑌)    &   𝑆 = (𝑁 ScMat 𝑃)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ((𝐶𝑀) · 1 ) ∈ 𝑆)

11.5.2  The characteristic factor function G

In this subsection the function 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛)))))))) is discussed. This function is involved in the representation of the product of the characteristic matrix of a given matrix and its adjunct as an infinite sum, see cpmadugsum 20502. 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 20490, cayhamlem3 20511 and cayhamlem4 20512.

Theoremfvmptnn04if 20473* The function values of a mapping from the nonnegative integers with four distinct cases. (Contributed by AV, 10-Nov-2019.)
𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵))))    &   (𝜑𝑆 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑌𝑉)    &   ((𝜑𝑁 = 0) → 𝑌 = 𝑁 / 𝑛𝐴)    &   ((𝜑 ∧ 0 < 𝑁𝑁 < 𝑆) → 𝑌 = 𝑁 / 𝑛𝐵)    &   ((𝜑𝑁 = 𝑆) → 𝑌 = 𝑁 / 𝑛𝐶)    &   ((𝜑𝑆 < 𝑁) → 𝑌 = 𝑁 / 𝑛𝐷)       (𝜑 → (𝐺𝑁) = 𝑌)

Theoremfvmptnn04ifa 20474* The function value of a mapping from the nonnegative integers with four distinct cases for the first case. (Contributed by AV, 10-Nov-2019.)
𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵))))    &   (𝜑𝑆 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ0)       ((𝜑𝑁 = 0 ∧ 𝑁 / 𝑛𝐴𝑉) → (𝐺𝑁) = 𝑁 / 𝑛𝐴)

Theoremfvmptnn04ifb 20475* The function value of a mapping from the nonnegative integers with four distinct cases for the second case. (Contributed by AV, 10-Nov-2019.)
𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵))))    &   (𝜑𝑆 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ0)       ((𝜑 ∧ (0 < 𝑁𝑁 < 𝑆) ∧ 𝑁 / 𝑛𝐵𝑉) → (𝐺𝑁) = 𝑁 / 𝑛𝐵)

Theoremfvmptnn04ifc 20476* The function value of a mapping from the nonnegative integers with four distinct cases for the third case. (Contributed by AV, 10-Nov-2019.)
𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵))))    &   (𝜑𝑆 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ0)       ((𝜑𝑁 = 𝑆𝑁 / 𝑛𝐶𝑉) → (𝐺𝑁) = 𝑁 / 𝑛𝐶)

Theoremfvmptnn04ifd 20477* The function value of a mapping from the nonnegative integers with four distinct cases for the forth case. (Contributed by AV, 10-Nov-2019.)
𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵))))    &   (𝜑𝑆 ∈ ℕ)    &   (𝜑𝑁 ∈ ℕ0)       ((𝜑𝑆 < 𝑁𝑁 / 𝑛𝐷𝑉) → (𝐺𝑁) = 𝑁 / 𝑛𝐷)

Theoremchfacfisf 20478* The "characteristic factor function" is a function from the nonnegative integers to polynomial matrices. (Contributed by AV, 8-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &    × = (.r𝑌)    &    = (-g𝑌)    &    0 = (0g𝑌)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝐺:ℕ0⟶(Base‘𝑌))

Theoremchfacfisfcpmat 20479* The "characteristic factor function" is a function from the nonnegative integers to constant polynomial matrices. (Contributed by AV, 19-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &    × = (.r𝑌)    &    = (-g𝑌)    &    0 = (0g𝑌)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))    &   𝑆 = (𝑁 ConstPolyMat 𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝐺:ℕ0𝑆)

Theoremchfacffsupp 20480* The "characteristic factor function" is finitely supported. (Contributed by AV, 20-Nov-2019.) (Proof shortened by AV, 23-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &    × = (.r𝑌)    &    = (-g𝑌)    &    0 = (0g𝑌)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → 𝐺 finSupp (0g𝑌))

Theoremchfacfscmulcl 20481* Closure of a scaled value of the "characteristic factor function". (Contributed by AV, 9-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &    × = (.r𝑌)    &    = (-g𝑌)    &    0 = (0g𝑌)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))    &   𝑋 = (var1𝑅)    &    · = ( ·𝑠𝑌)    &    = (.g‘(mulGrp‘𝑃))       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝐾 ∈ ℕ0) → ((𝐾 𝑋) · (𝐺𝐾)) ∈ (Base‘𝑌))

Theoremchfacfscmul0 20482* A scaled value of the "characteristic factor function" is zero almost always. (Contributed by AV, 9-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &    × = (.r𝑌)    &    = (-g𝑌)    &    0 = (0g𝑌)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))    &   𝑋 = (var1𝑅)    &    · = ( ·𝑠𝑌)    &    = (.g‘(mulGrp‘𝑃))       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝐾 ∈ (ℤ‘(𝑠 + 2))) → ((𝐾 𝑋) · (𝐺𝐾)) = 0 )

Theoremchfacfscmulfsupp 20483* A mapping of scaled values of the "characteristic factor function" is finitely supported. (Contributed by AV, 8-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &    × = (.r𝑌)    &    = (-g𝑌)    &    0 = (0g𝑌)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))    &   𝑋 = (var1𝑅)    &    · = ( ·𝑠𝑌)    &    = (.g‘(mulGrp‘𝑃))       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑖 ∈ ℕ0 ↦ ((𝑖 𝑋) · (𝐺𝑖))) finSupp 0 )

Theoremchfacfscmulgsum 20484* Breaking up a sum of values of the "characteristic factor function" scaled by a polynomial. (Contributed by AV, 9-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &    × = (.r𝑌)    &    = (-g𝑌)    &    0 = (0g𝑌)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))    &   𝑋 = (var1𝑅)    &    · = ( ·𝑠𝑌)    &    = (.g‘(mulGrp‘𝑃))    &    + = (+g𝑌)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 𝑋) · (𝐺𝑖)))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))))) + ((((𝑠 + 1) 𝑋) · (𝑇‘(𝑏𝑠))) ((𝑇𝑀) × (𝑇‘(𝑏‘0))))))

Theoremchfacfpmmulcl 20485* Closure of the value of the "characteristic factor function" multiplied with a constant polynomial matrix. (Contributed by AV, 23-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &    × = (.r𝑌)    &    = (-g𝑌)    &    0 = (0g𝑌)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))    &    = (.g‘(mulGrp‘𝑌))       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝐾 ∈ ℕ0) → ((𝐾 (𝑇𝑀)) × (𝐺𝐾)) ∈ (Base‘𝑌))

Theoremchfacfpmmul0 20486* The value of the "characteristic factor function" multiplied with a constant polynomial matrix is zero almost always. (Contributed by AV, 23-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &    × = (.r𝑌)    &    = (-g𝑌)    &    0 = (0g𝑌)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))    &    = (.g‘(mulGrp‘𝑌))       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠))) ∧ 𝐾 ∈ (ℤ‘(𝑠 + 2))) → ((𝐾 (𝑇𝑀)) × (𝐺𝐾)) = 0 )

Theoremchfacfpmmulfsupp 20487* A mapping of values of the "characteristic factor function" multiplied with a constant polynomial matrix is finitely supported. (Contributed by AV, 23-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &    × = (.r𝑌)    &    = (-g𝑌)    &    0 = (0g𝑌)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))    &    = (.g‘(mulGrp‘𝑌))       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑖 ∈ ℕ0 ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖))) finSupp 0 )

Theoremchfacfpmmulgsum 20488* Breaking up a sum of values of the "characteristic factor function" multiplied with a constant polynomial matrix. (Contributed by AV, 23-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &    × = (.r𝑌)    &    = (-g𝑌)    &    0 = (0g𝑌)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))    &    = (.g‘(mulGrp‘𝑌))    &    + = (+g𝑌)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 (𝑇𝑀)) × ((𝑇‘(𝑏‘(𝑖 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑖))))))) + ((((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑠))) ((𝑇𝑀) × (𝑇‘(𝑏‘0))))))

Theoremchfacfpmmulgsum2 20489* Breaking up a sum of values of the "characteristic factor function" multiplied with a constant polynomial matrix. (Contributed by AV, 23-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &    × = (.r𝑌)    &    = (-g𝑌)    &    0 = (0g𝑌)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))    &    = (.g‘(mulGrp‘𝑌))    &    + = (+g𝑌)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ (((𝑖 (𝑇𝑀)) × (𝑇‘(𝑏‘(𝑖 − 1)))) (((𝑖 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑖)))))) + ((((𝑠 + 1) (𝑇𝑀)) × (𝑇‘(𝑏𝑠))) ((𝑇𝑀) × (𝑇‘(𝑏‘0))))))

Theoremcayhamlem1 20490* Lemma 1 for cayleyhamilton 20514. (Contributed by AV, 11-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &    × = (.r𝑌)    &    = (-g𝑌)    &    0 = (0g𝑌)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 ((𝑇𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) ((𝑇𝑀) × (𝑇‘(𝑏𝑛))))))))    &    = (.g‘(mulGrp‘𝑌))       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 (𝑇𝑀)) × (𝐺𝑖)))) = 0 )

11.5.3  The Cayley-Hamilton theorem

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

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

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

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

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

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

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

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

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

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

Writing

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

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

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

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

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

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

(8) 0 = p(A) .

This completes the proof."

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

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

p(A) = 0.

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

Actually, the definition of the characteristic polynomial of a square matrix requires some attention. According to df-chpmat 20451, the characteristic polynomial of an 𝑁 x 𝑁 matrix 𝑀 over a ring 𝑅 is defined as

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

where 𝐷 = (𝑁 maDet 𝑃) is the function mapping an 𝑁 x 𝑁 matrix over the polynomial ring over the ring 𝑅 to its determinant, 𝑋 = (var1𝑅) is the variable of the polynomials over 𝑅, 1 is the 𝑁 x 𝑁 identity matrix as matrix over the polynomial ring over the ring 𝑅 (not the 𝑁 x 𝑁 identity matrix of the matrices over the ring 𝑅!) and (𝑇𝑀) = ((𝑁 matToPolyMat 𝑅)‘𝑀) is the matrix 𝑀 over a ring 𝑅 transformed into a constant matrix over the polynomial ring over the ring 𝑅. Thus · is the multiplication of a polynomial matrix with a "scalar", i.e. a polynomial (see chpmatval 20455).

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

After the preliminaries are clarified, the objective of the Cayley-Hamilton theorem must be considered. As described in Wikipedia, the matrix 𝑀 must be "inserted" into its characteristic polynomial in an appropriate way. Since every polynomial can be represented as infinite, but finitely supported sum of monomials scaled by the corresponding coefficients (see ply1coe 19487), also the characteristic polynomial can be written in this way:

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

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

By this, we can "define" the insertion of the matrix M into its characteristic polynomial by "p(M) = sum( pi * M^i)", see also cayleyhamilton1 20516. Here, * is the scalar multiplication in the algebra of the matrices over the ring 𝑅.

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

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

class variable/ informal identifier definiens explanation
𝑁 An arbitrary finite set, used as dimension for matrices
𝑅 An arbitrary (commutative) ring: 𝑅 ∈ CRing
B(R) (Base‘𝑅) Base set of (the ring) 𝑅
𝐴 (𝑁 Mat 𝑅) Algebra of 𝑁 x 𝑁 matrices over (the ring) 𝑅
𝐵 (Base‘𝐴) Base set of the algebra of 𝑁 x 𝑁 matrices, i .e. the set of all 𝑁 x 𝑁 matrices
𝑀 An arbitrary 𝑁 x 𝑁 matrix
𝑃 (Poly1𝑅) The algebra of polynomials over (the ring) 𝑅
B(P) (Base‘𝑃) Base set of the algebra of polynomials, i .e. the set of all polynomials
𝑋, XR (var1𝑅) The variable of polynomials over (the ring) 𝑅
𝑌, XA (var1𝐴) The variable of polynomials over matrices of the algebra 𝐴
(.g‘(mulGrp‘𝑃)) The group exponentiation for polynomials over (the ring) 𝑅
^ Arbitrary group exponentiation
𝑈 (algSc‘𝑃) The injection of scalars, i.e. elements of (the ring) 𝑅 into the base set of the algebra of polynomials over 𝑅
(𝑈𝑝), S(p) ((algSc‘𝑃)‘𝑝) The element 𝑝 of (the ring) 𝑅 represented as polynomial over 𝑅
𝑌 (𝑁 Mat 𝑃) Algebra of 𝑁 x 𝑁 matrices over (the polynomial ring) 𝑃 over the ring 𝑅
B(Y) (Base‘𝑌) Base set of the algebra of polynomial 𝑁 x 𝑁 matrices, i .e. the set of all polynomial 𝑁 x 𝑁 matrices
𝑄 (Poly1𝐴) Algebra of polynomials over the ring of 𝑁 x 𝑁 matrices over the ring 𝑅
B(Q) (Base‘𝑄) Base set of the algebra of polynomials over the ring of 𝑁 x 𝑁 matrices over the ring 𝑅, i .e. the set of all polynomials having 𝑁 x 𝑁 matrices as coefficients
+, + (+g𝑌) The addition of polynomial matrices
, - (-g𝑌) The subtraction of polynomial matrices
·, *Y ( ·𝑠𝑌) The multiplication of a polynomial matrix with a scalar ( i. e. a polynomial)
*A ( ·𝑠𝐴) The multiplication of a matrix with a scalar ( i. e. an element of the underlying ring)
*Q ( ·𝑠𝑄) The multiplication of a polynomial over matrices with a scalar ( i. e. a matrix)
×, xY (.r𝑌) The multiplication of polynomial matrices
xA (.r𝐴) The multiplication of matrices
xQ (.r𝑄) The multiplication of polynomials over matrices
1, 1Y (1r𝑌) The identity matrix in the algebra of polynomial matrices over 𝑅
1A (1r𝐴) The identity matrix in the algebra of matrices over 𝑅
0, 0Y (0g𝑌) The zero matrix in the algebra of matrices consisting of polynomials
𝑇 (𝑁 matToPolyMat 𝑅) The transformation of an 𝑁 x 𝑁 matrix over 𝑅 into a polynomial 𝑁 x 𝑁 matrix over 𝑅
T1(M) (𝑇𝑀) The matrix M transformed into a polynomial 𝑁 x 𝑁 matrix over 𝑅
U(M) (𝑈𝑀) The (constant) polynomial 𝑁 x 𝑁 matrix M transformed into a matrix over the ring 𝑅. Inverse function of 𝑇: (𝑇‘(𝑈𝑀)) = 𝑀 (see m2cpminvid2 20379 )
T2(M) ((𝑁 pMatToMatPoly 𝑅)‘𝑀) The polynomial 𝑁 x 𝑁 matrix M transformed into a polynomial over the 𝑁 x 𝑁 matrices over 𝑅
𝐼, c(M) ((𝑋 · 1 ) (𝑇𝑀)) The characteristic matrix of a matrix 𝑀, i.e. the matrix whose determinant is the characteristic polynomial of the matrix 𝑀
𝐶 (𝑁 CharPlyMat 𝑅) The function mapping a matrix over (a ring) 𝑅 to its characteristic polynomial
𝐾, p(M) (𝐶𝑀) The characteristic polynomial of a matrix over (a ring) 𝑅
𝐻 (𝐾 · 1 ) The scalar matrix (diagonal matrix) with the characteristic polynomial of a matrix as diagional elements
𝐽 (𝑁 maAdju 𝑃) The function mapping a matrix consisting of polynomials to its adjugate ("matrix of cofactors")
𝑊, adj(cm(M)) (𝐽𝐼) The adjugate of the characteristic matrix of the matrix 𝑀

Using this notation, we have:
1. "c(M) e. B(Y)", or 𝐼 ∈ (Base‘𝑌), see chmatcl 20452
2. "p(M) e. B(P)", or 𝐾 ∈ (Base‘𝑃), see chpmatply1 20456
3. "T(M) e. B(Y)", or (𝑇𝑀) ∈ (Base‘𝑌), see mat2pmatbas 20350
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 20412:
adj(cm(M)) = sumi=0 to s ( XR ^i *Y T1(b(i)) )
where b(i) are matrices over the ring 𝑅, so T1(b(i)) are constant polynomial matrices.
This step corresponds to (3) in Wikipedia. In contrast to Wikipedia, we write 𝑊 as finite sum of not exactly determined number of summands, which may be greater than needed (including summands of value 0). This will be sufficient to provide a representation of (𝐼 × 𝑊) as infinite, but finitely supported sum, see step 3.
2. Write (𝐼 × 𝑊), the product of the characteristic matrix and its adjugate as finite sum of scaled monomials, see cpmadugsumfi 20501. This representation is obtained by replacing 𝑊 by the representation resulting from step 1. and performing calculation rules available for the associative algebra of matrices over polynomials over a commutative ring:
cm(M) *Y adj(cm(M)) = sumi=0 to s ( XR ^i *Y ( T1(b(i-1)) - T1(M) xY T1(b(i)) ) ) + XR ^(s+1) *Y ( T1(b(s)) - T1(M) xY T1(b(0))
where b(i) are matrices over 𝑅, so T1(b(i)) are constant polynomial matrices:
= cm(M) *Y sumi=0 to s ( XR ^i *Y T1(b(i)) ) [see pmatcollpw3fi1 20412 (step 1.)]
= ( ( XA *Y 1Y ) - T1(M) ) *Y sumi=0 to s ( XR ^i *Y T1(b(i)) ) [def. of cm(M)]
= ( XA *Y 1Y ) *Y sumi=0 to s ( XR ^i *Y T1(b(i)) ) - T1(M) *Y sumi=0 to s ( XR ^i *Y T1(b(i)) ) [see rngsubdir 18423]
= sumi=0 to s ( XR ^i *Y ( T1(b(i-1)) - T1(M) xY T1(b(i)) ) ) + XR ^(s+1) *Y ( T1(b(s)) - T1(M) xY T1(b(0)) [see cpmadugsumlemF 20500]
This step corresponds partially to (4) in Wikipedia.
3. Write (𝐼 × 𝑊) as infinite, but finitely supported sum of scaled monomials, see cpmadugsum 20502:
cm(M) * adj(cm(M)) = sumi ( XR ^i *Y G(i) )
This representation is obtained by defining a function G for the coefficients, which we call "characteristic factor function", see chfacfisf 20478, which covers the special terms and the padding with 0. G(i) is a constant polynomial matrix (see chfacfisfcpmat 20479). This step corresponds partially to (4) in Wikipedia, with summands of value 0 added.
4. Write 𝐻 = (𝐾 · 1 ), the scalar matrix (diagonal matrix) with the characteristic polynomial of a matrix as diagional elements, as infinite, but finitely supported sum of scaled monomials. See cpmidgsum 20492:
p(m) *Y IY = sumi ( XR ^i *Y ( S(pi) *Y IY ) )
The proof of cpmidgsum 20492 is making use of pmatcollpwscmat 20415, because 𝐻 = (𝐾 · 1 ) is a scalar/diagonal polynomial matrix with the characteristic polynomial "p(M)" as diagonal entries (since pi is an element of the ring 𝑅, S(pi) is a (constant) polynomial). This corresponds to (5) in Wikipedia, with summands of value 0 added.
5. Transform the sum representation of (𝐼 × 𝑊) from step 3. into polynomials over matrices:
T2(cm(M) * adj(cm(M))) = sumi ( U(G(i)) *Q XA ^i ) [see cpmadumatpoly 20507]
where U(G(i)) is a matrix over the ring 𝑅.
6. Transform the sum representation of 𝐻 from step 4. into polynomials over matrices:
T2(p(m) *Y IY) = sumi ( pi *A IA ) *Q XA ^i ) [see cpmidpmat 20497]
7. Equate the sum representations resulting from steps 5. and 6. by using cpmadurid 20491 to obtain the equation
sumi ( U(G(i)) *Q XA ^i ) = sumi ( pi *A IA ) *Q XA ^i ):
sumi ( U(G(i)) *Q XA ^i )
= T2(cm(M) * adj(cm(M))) [see step 5.]
= T2(p(m) *Y IY) [see cpmadurid 20491]
= sumi ( pi *A IA ) *Q XA ^i ) [see step 6.]
Note that this step is contained in the proof of chcoeffeq 20510, see step 9. This step corresponds to the conclusion from (4) and (5) in Wikipedia, with summands of value 0 added.
8. Compare the sum representations of step 7. to obtain the equations U(G(i)) = pi *A IA , see chcoeffeqlem 20509. 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 20510:
sumi ( M^i xA U(G(i)) ) = sumi ( M^i xA ( pi *A IA) )
This corresponds to (7) in Wikipedia.
10. Transform the right hand side of the equation in step 9. into an appropriate form, see cayhamlem3 20511:
sumi ( pi *A M^i )
= sumi ( M^i xA ( pi *A IA) ) [see cayhamlem2 20508]
= sumi ( M^i xA U(G(i)) ) [see chcoeffeq 20510]
11. Apply the theorem for telescoping sums, see telgsumfz 18210, to the sum sumi ( T1(M)^i xY G(i) ), which results in an equation to 0:
sumi ( T1(M)^i xY G(i) ) = 0Y, see cayhamlem1 20490:
sumi ( T1(M)^i xY G(i) )
= sumi=1 to s ( T1(M)^i xY T1(b(i-1)) - T1(M)^(i+1) xY T1(b(i)) )
+ ( T1(M)^(s+1) xY T1(b(s)) - T1(M) xY T1(b(0)) ) [see chfacfpmmulgsum2 20489]
= ( T1(M) xY T1(b(0)) - T1(M)^(s+1) xY T1(b(s)) ) + ( T1 M)^(s+1) xY T1(b(s)) - T1(M) xY T1(b(0)) ) [see telgsumfz 18210]
= 0Y [see grpnpncan0 17334] This step corresponds partially to (8) in Wikipedia.
12. Since 𝑇 is a ring homomorphism (see mat2pmatrhm 20358), 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 20512:
sumi ( pi *A M^i )
= sumi ( M^i xA U(G(i)) ) [see cayhamlem3 20511 (step 10.)]
= U(T1(sumi ( M^i xA U(G(i)) ))) [see m2cpminvid 20377]
= U(sumi T1( M^i xA U(G(i)) )) [see gsummptmhm 18163]
= U(sumi ( T1(M^i) xY T1(U(G(i))) )) [see rhmmul 18550]
= U(sumi ( T1(M)^i xY T1(U(G(i))) )) [see mhmmulg 17406]
= U(sumi ( T1(M)^i xY G(i) )) [see m2cpminvid2 20379 ]
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 20353 and mat2pmatrhm 20358) to transform the equality resulting from steps 11. and 12. into the desired equation sumi ( pi *A M^i ) = 0A , see cayleyhamilton 20514 resp. cayleyhamilton0 20513:
sumi ( pi *A M^i )
= U(sumi ( T1(M)^i xY G(i) )) [see cayhamlem4 20512 (step 12.)]
= U(0Y ) [see cayhamlem1 20490 (step 11.)]
= 0A [see m2cpminv0 20385]
The transformations in steps 5., 6., 10., 12. and 13. are not mentioned in the proof provided in Wikipedia, since it makes no distinction between a matrix over a ring 𝑅 and its representation as matrix over the polynomial ring over the ring 𝑅 in general!

Theoremcpmadurid 20491 The right-hand fundamental relation of the adjugate (see madurid 20269) applied to the characteristic matrix of a matrix. (Contributed by AV, 25-Oct-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝑋 = (var1𝑅)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &    = (-g𝑌)    &    · = ( ·𝑠𝑌)    &    1 = (1r𝑌)    &   𝐼 = ((𝑋 · 1 ) (𝑇𝑀))    &   𝐽 = (𝑁 maAdju 𝑃)    &    × = (.r𝑌)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝐼 × (𝐽𝐼)) = ((𝐶𝑀) · 1 ))

Theoremcpmidgsum 20492* Representation of the identity matrix multiplied with the characteristic polynomial of a matrix as group sum. (Contributed by AV, 7-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝑋 = (var1𝑅)    &    = (.g‘(mulGrp‘𝑃))    &    · = ( ·𝑠𝑌)    &    1 = (1r𝑌)    &   𝑈 = (algSc‘𝑃)    &   𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝐾 = (𝐶𝑀)    &   𝐻 = (𝐾 · 1 )       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐻 = (𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 𝑋) · ((𝑈‘((coe1𝐾)‘𝑛)) · 1 )))))

Theoremcpmidgsumm2pm 20493* Representation of the identity matrix multiplied with the characteristic polynomial of a matrix as group sum with a matrix to polynomial matrix transformation. (Contributed by AV, 13-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝑋 = (var1𝑅)    &    = (.g‘(mulGrp‘𝑃))    &    · = ( ·𝑠𝑌)    &    1 = (1r𝑌)    &   𝑈 = (algSc‘𝑃)    &   𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝐾 = (𝐶𝑀)    &   𝐻 = (𝐾 · 1 )    &   𝑂 = (1r𝐴)    &    = ( ·𝑠𝐴)    &   𝑇 = (𝑁 matToPolyMat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐻 = (𝑌 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 𝑋) · (𝑇‘(((coe1𝐾)‘𝑛) 𝑂))))))

Theoremcpmidpmatlem1 20494* Lemma 1 for cpmidpmat 20497. (Contributed by AV, 13-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝑋 = (var1𝑅)    &    = (.g‘(mulGrp‘𝑃))    &    · = ( ·𝑠𝑌)    &    1 = (1r𝑌)    &   𝑈 = (algSc‘𝑃)    &   𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝐾 = (𝐶𝑀)    &   𝐻 = (𝐾 · 1 )    &   𝑂 = (1r𝐴)    &    = ( ·𝑠𝐴)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐺 = (𝑘 ∈ ℕ0 ↦ (((coe1𝐾)‘𝑘) 𝑂))       (𝐿 ∈ ℕ0 → (𝐺𝐿) = (((coe1𝐾)‘𝐿) 𝑂))

Theoremcpmidpmatlem2 20495* Lemma 2 for cpmidpmat 20497. (Contributed by AV, 14-Nov-2019.) (Proof shortened by AV, 7-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝑋 = (var1𝑅)    &    = (.g‘(mulGrp‘𝑃))    &    · = ( ·𝑠𝑌)    &    1 = (1r𝑌)    &   𝑈 = (algSc‘𝑃)    &   𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝐾 = (𝐶𝑀)    &   𝐻 = (𝐾 · 1 )    &   𝑂 = (1r𝐴)    &    = ( ·𝑠𝐴)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐺 = (𝑘 ∈ ℕ0 ↦ (((coe1𝐾)‘𝑘) 𝑂))       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐺 ∈ (𝐵𝑚0))

Theoremcpmidpmatlem3 20496* Lemma 3 for cpmidpmat 20497. (Contributed by AV, 14-Nov-2019.) (Proof shortened by AV, 7-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝑋 = (var1𝑅)    &    = (.g‘(mulGrp‘𝑃))    &    · = ( ·𝑠𝑌)    &    1 = (1r𝑌)    &   𝑈 = (algSc‘𝑃)    &   𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝐾 = (𝐶𝑀)    &   𝐻 = (𝐾 · 1 )    &   𝑂 = (1r𝐴)    &    = ( ·𝑠𝐴)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐺 = (𝑘 ∈ ℕ0 ↦ (((coe1𝐾)‘𝑘) 𝑂))       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝐺 finSupp (0g𝐴))

Theoremcpmidpmat 20497* Representation of the identity matrix multiplied with the characteristic polynomial of a matrix as polynomial over the ring of matrices. (Contributed by AV, 14-Nov-2019.) (Revised by AV, 7-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝑋 = (var1𝑅)    &    = (.g‘(mulGrp‘𝑃))    &    · = ( ·𝑠𝑌)    &    1 = (1r𝑌)    &   𝑈 = (algSc‘𝑃)    &   𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝐾 = (𝐶𝑀)    &   𝐻 = (𝐾 · 1 )    &   𝑂 = (1r𝐴)    &    = ( ·𝑠𝐴)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝑊 = (Base‘𝑌)    &   𝑄 = (Poly1𝐴)    &   𝑍 = (var1𝐴)    &    = ( ·𝑠𝑄)    &   𝐸 = (.g‘(mulGrp‘𝑄))    &   𝐼 = (𝑁 pMatToMatPoly 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝐼𝐻) = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ ((((coe1𝐾)‘𝑛) 𝑂) (𝑛𝐸𝑍)))))