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Theorem List for Metamath Proof Explorer - 20301-20400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempmatcollpw1lem1 20301* Lemma 1 for pmatcollpw1 20303. (Contributed by AV, 28-Sep-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) ∧ 𝐼𝑁𝐽𝑁) → (𝑛 ∈ ℕ0 ↦ ((𝐼(𝑀 decompPMat 𝑛)𝐽) × (𝑛 𝑋))) finSupp (0g𝑃))
 
Theorempmatcollpw1lem2 20302* Lemma 2 for pmatcollpw1 20303: An entry of a polynomial matrix is the sum of the entries of the matrix consisting of the coefficients in the entries of the polynomial matrix multiplied with the corresponding power of the variable. (Contributed by AV, 25-Sep-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) ∧ (𝑎𝑁𝑏𝑁)) → (𝑎𝑀𝑏) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝑎(𝑀 decompPMat 𝑛)𝑏) × (𝑛 𝑋)))))
 
Theorempmatcollpw1 20303* Write a polynomial matrix as a matrix of sums of scaled monomials. (Contributed by AV, 29-Sep-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 = (𝑖𝑁, 𝑗𝑁 ↦ (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))))))
 
Theorempmatcollpw2lem 20304* Lemma for pmatcollpw2 20305. (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 20305* 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 20306 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 20297 (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 20307 Lemma for pmatcollpw 20308. (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 20308* 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 20309* 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 20310* Lemma for pmatcollpw3 20311 and pmatcollpw3fi 20312: 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 20311* 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 20312* 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 20313* Lemma 1 for pmatcollpw3fi1 20315. (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 20314* Lemma 2 for pmatcollpw3fi1 20315. (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 20315* 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 20316 Lemma 1 for pmatcollpwscmat 20318. (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 20317 Lemma 2 for pmatcollpwscmat 20318. (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 20318* 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 20350, 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 20342 and pm2mprngiso 20349), 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 20338).

In this section, the following conventions are mostly used:

  • 𝑅 is a (unital) ring (see df-ring 18279)
  • 𝑃 = (Poly1𝑅) is the polynomial algebra over (the ring) 𝑅 (see df-ply1 19277)
    • 𝐾 = (Base‘𝑃) is its base set (see df-base 15584)
    • 𝑌 = (var1𝑅) is its variable (see df-vr1 19276)
    • · = ( ·𝑠𝑃) is its scalar multiplication (see df-vsca 15669 or lmodvscl 18610)
    • 𝐸 = (.g‘(mulGrp‘𝑃)) is its exponentiation (see df-mulg 17256)
  • 𝐴 = (𝑁 Mat 𝑅) is the algebra of N x N matrices over (the ring) 𝑅 (see df-mat 19936)
  • 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 20319 Extend class notation with the transformation of a polynomial matrix into a polynomial over matrices.
class pMatToMatPoly
 
Definitiondf-pm2mp 20320* 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 20321* Lemma for pm2mpf1 20326. (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 20322* 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 20323* 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 20324 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 20325 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 20326 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 20327 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 20328 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 20329* 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 20330* Lemma for mply1topmatcl 20332. (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 20331* 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 20338). (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 20332* 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 20333* Lemma 1 for mp2pm2mp 20338. (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 20334* Lemma 2 for mp2pm2mp 20338. (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 20335* Lemma 3 for mp2pm2mp 20338. (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 20336* Lemma 4 for mp2pm2mp 20338. (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 20337* Lemma 5 for mp2pm2mp 20338. (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 20338* 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 20339* Lemma 2 for pm2mpghm 20343. (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 20340 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 20341 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 20342 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 20343 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 20344 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 20345* Lemma 1 for pm2mpmhm 20347. (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 20346* Lemma 2 for pm2mpmhm 20347. (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 20347 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 20348 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 20349 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 20350 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 20351 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 20352* 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 20354) 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 20353 Extend class notation with the characteristic polynomial.
class CharPlyMat
 
Definitiondf-chpmat 20354* 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 20355 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 20356 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 20357* 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 20358 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 20359 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 20360* 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 20361 The characteristic polynomial of the empty matrix. (Contributed by AV, 6-Aug-2019.)
𝐶 = (∅ CharPlyMat 𝑅)       (𝑅 ∈ Ring → (𝐶‘∅) = (1r‘(Poly1𝑅)))
 
Theoremchpmat1dlem 20362 Lemma for chpmat1d 20363. (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 20363 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 20364 Lemma 0 for chpdmat 20368. (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 20365 Lemma 1 for chpdmat 20368. (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 20366 Lemma 2 for chpdmat 20368. (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 20367 Lemma 3 for chpdmat 20368. (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 20368* 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 20369* 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 20370* 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 20371* 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 20372* 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 20373 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 20374 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 20375 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 20405. 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 20393, cayhamlem3 20414 and cayhamlem4 20415.

 
Theoremfvmptnn04if 20376* 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 20377* 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 20378* 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 20379* 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 20380* 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 20381* 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 20382* 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 20383* 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 20384* 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 20385* 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 20386* 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 20387* 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 20388* 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 20389* 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 20390* 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 20391* 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 20392* 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 20393* Lemma 1 for cayleyhamilton 20417. (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 20354, 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 20358).

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 19391), 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 20419. 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 20282 )
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 20355
  2. "p(M) e. B(P)", or 𝐾 ∈ (Base‘𝑃), see chpmatply1 20359
  3. "T(M) e. B(Y)", or (𝑇𝑀) ∈ (Base‘𝑌), see mat2pmatbas 20253
  4. 𝐽:(Base‘𝑌)⟶(Base‘𝑌), see maduf 20169
  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 20315:
    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 20404. This representation is obtained by replacing 𝑊 by the representation resulting from step 1. and performing calculation rules available for the associative algebra of matrices over polynomials over a commutative ring:
    cm(M) *Y adj(cm(M)) = sumi=0 to s ( XR ^i *Y ( T1(b(i-1)) - T1(M) xY T1(b(i)) ) ) + XR ^(s+1) *Y ( T1(b(s)) - T1(M) xY T1(b(0))
    where b(i) are matrices over 𝑅, so T1(b(i)) are constant polynomial matrices:
    cm(M) *Y adj(cm(M))
    = cm(M) *Y sumi=0 to s ( XR ^i *Y T1(b(i)) ) [see pmatcollpw3fi1 20315 (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 18330]
    = 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 20403]
    This step corresponds partially to (4) in Wikipedia.
  3. Write (𝐼 × 𝑊) as infinite, but finitely supported sum of scaled monomials, see cpmadugsum 20405:
    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 20381, which covers the special terms and the padding with 0. G(i) is a constant polynomial matrix (see chfacfisfcpmat 20382). 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 20395:
    p(m) *Y IY = sumi ( XR ^i *Y ( S(pi) *Y IY ) )
    The proof of cpmidgsum 20395 is making use of pmatcollpwscmat 20318, 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 20410]
    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 20400]
  7. Equate the sum representations resulting from steps 5. and 6. by using cpmadurid 20394 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 20394]
    = sumi ( pi *A IA ) *Q XA ^i ) [see step 6.]
    Note that this step is contained in the proof of chcoeffeq 20413, 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 20412. 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 20413:
    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 20414:
    sumi ( pi *A M^i )
    = sumi ( M^i xA ( pi *A IA) ) [see cayhamlem2 20411]
    = sumi ( M^i xA U(G(i)) ) [see chcoeffeq 20413]
  11. Apply the theorem for telescoping sums, see telgsumfz 18117, 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 20393:
    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 20392]
    = ( 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 18117]
    = 0Y [see grpnpncan0 17226] This step corresponds partially to (8) in Wikipedia.
  12. Since 𝑇 is a ring homomorphism (see mat2pmatrhm 20261), 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 20415:
    sumi ( pi *A M^i )
    = sumi ( M^i xA U(G(i)) ) [see cayhamlem3 20414 (step 10.)]
    = U(T1(sumi ( M^i xA U(G(i)) ))) [see m2cpminvid 20280]
    = U(sumi T1( M^i xA U(G(i)) )) [see gsummptmhm 18070]
    = U(sumi ( T1(M^i) xY T1(U(G(i))) )) [see rhmmul 18457]
    = U(sumi ( T1(M)^i xY T1(U(G(i))) )) [see mhmmulg 17298]
    = U(sumi ( T1(M)^i xY G(i) )) [see m2cpminvid2 20282 ]
  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 20256 and mat2pmatrhm 20261) to transform the equality resulting from steps 11. and 12. into the desired equation sumi ( pi *A M^i ) = 0A , see cayleyhamilton 20417 resp. cayleyhamilton0 20416:
    sumi ( pi *A M^i )
    = U(sumi ( T1(M)^i xY G(i) )) [see cayhamlem4 20415 (step 12.)]
    = U(0Y ) [see cayhamlem1 20393 (step 11.)]
    = 0A [see m2cpminv0 20288]
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 20394 The right-hand fundamental relation of the adjugate (see madurid 20172) 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 20395* 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 20396* 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 20397* Lemma 1 for cpmidpmat 20400. (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 20398* Lemma 2 for cpmidpmat 20400. (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 20399* Lemma 3 for cpmidpmat 20400. (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 20400* 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)))))
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