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Theorem List for Metamath Proof Explorer - 21301-21400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-decpmat 21301* Define the decomposition of polynomial matrices. This function collects the coefficients of a polynomial matrix 𝑚 belong to the 𝑘 th power of the polynomial variable for each entry of 𝑚. (Contributed by AV, 2-Dec-2019.)
decompPMat = (𝑚 ∈ V, 𝑘 ∈ ℕ0 ↦ (𝑖 ∈ dom dom 𝑚, 𝑗 ∈ dom dom 𝑚 ↦ ((coe1‘(𝑖𝑚𝑗))‘𝑘)))
 
Theoremdecpmatval0 21302* The matrix consisting of the coefficients in the polynomial entries of a polynomial matrix for the same power, most general version. (Contributed by AV, 2-Dec-2019.)
((𝑀𝑉𝐾 ∈ ℕ0) → (𝑀 decompPMat 𝐾) = (𝑖 ∈ dom dom 𝑀, 𝑗 ∈ dom dom 𝑀 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝐾)))
 
Theoremdecpmatval 21303* The matrix consisting of the coefficients in the polynomial entries of a polynomial matrix for the same power, general version for arbitrary matrices. (Contributed by AV, 28-Sep-2019.) (Revised by AV, 2-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)       ((𝑀𝐵𝐾 ∈ ℕ0) → (𝑀 decompPMat 𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝐾)))
 
Theoremdecpmate 21304 An entry of the matrix consisting of the coefficients in the entries of a polynomial matrix is the corresponding coefficient in the polynomial entry of the given matrix. (Contributed by AV, 28-Sep-2019.) (Revised by AV, 2-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)       (((𝑅𝑉𝑀𝐵𝐾 ∈ ℕ0) ∧ (𝐼𝑁𝐽𝑁)) → (𝐼(𝑀 decompPMat 𝐾)𝐽) = ((coe1‘(𝐼𝑀𝐽))‘𝐾))
 
Theoremdecpmatcl 21305 Closure of the decomposition of a polynomial matrix: The matrix consisting of the coefficients in the polynomial entries of a polynomial matrix for the same power is a matrix. (Contributed by AV, 28-Sep-2019.) (Revised by AV, 2-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐷 = (Base‘𝐴)       ((𝑅𝑉𝑀𝐵𝐾 ∈ ℕ0) → (𝑀 decompPMat 𝐾) ∈ 𝐷)
 
Theoremdecpmataa0 21306* The matrix consisting of the coefficients in the polynomial entries of a polynomial matrix for the same power is 0 for almost all powers. (Contributed by AV, 3-Nov-2019.) (Revised by AV, 3-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &   𝐴 = (𝑁 Mat 𝑅)    &    0 = (0g𝐴)       ((𝑅 ∈ Ring ∧ 𝑀𝐵) → ∃𝑠 ∈ ℕ0𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝑀 decompPMat 𝑥) = 0 ))
 
Theoremdecpmatfsupp 21307* The mapping to the matrices consisting of the coefficients in the polynomial entries of a given matrix for the same power is finitely supported. (Contributed by AV, 5-Oct-2019.) (Revised by AV, 3-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &   𝐴 = (𝑁 Mat 𝑅)    &    0 = (0g𝐴)       ((𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑘 ∈ ℕ0 ↦ (𝑀 decompPMat 𝑘)) finSupp 0 )
 
Theoremdecpmatid 21308 The matrix consisting of the coefficients in the polynomial entries of the identity matrix is an identity or a zero matrix. (Contributed by AV, 28-Sep-2019.) (Revised by AV, 2-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐼 = (1r𝐶)    &   𝐴 = (𝑁 Mat 𝑅)    &    0 = (0g𝐴)    &    1 = (1r𝐴)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (𝐼 decompPMat 𝐾) = if(𝐾 = 0, 1 , 0 ))
 
Theoremdecpmatmullem 21309* Lemma for decpmatmul 21310. (Contributed by AV, 20-Oct-2019.) (Revised by AV, 3-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑈𝐵𝑊𝐵) ∧ (𝐼𝑁𝐽𝑁𝐾 ∈ ℕ0)) → (𝐼((𝑈(.r𝐶)𝑊) decompPMat 𝐾)𝐽) = (𝑅 Σg (𝑡𝑁 ↦ (𝑅 Σg (𝑙 ∈ (0...𝐾) ↦ (((coe1‘(𝐼𝑈𝑡))‘𝑙)(.r𝑅)((coe1‘(𝑡𝑊𝐽))‘(𝐾𝑙))))))))
 
Theoremdecpmatmul 21310* The matrix consisting of the coefficients in the polynomial entries of the product of two polynomial matrices is a sum of products of the matrices consisting of the coefficients in the polynomial entries of the polynomial matrices for the same power. (Contributed by AV, 21-Oct-2019.) (Revised by AV, 3-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &   𝐴 = (𝑁 Mat 𝑅)       ((𝑅 ∈ Ring ∧ (𝑈𝐵𝑊𝐵) ∧ 𝐾 ∈ ℕ0) → ((𝑈(.r𝐶)𝑊) decompPMat 𝐾) = (𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(.r𝐴)(𝑊 decompPMat (𝐾𝑘))))))
 
Theoremdecpmatmulsumfsupp 21311* Lemma 0 for pm2mpmhm 21358. (Contributed by AV, 21-Oct-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &   𝐴 = (𝑁 Mat 𝑅)    &    · = (.r𝐴)    &    0 = (0g𝐴)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑙 ∈ ℕ0 ↦ (𝐴 Σg (𝑘 ∈ (0...𝑙) ↦ ((𝑥 decompPMat 𝑘) · (𝑦 decompPMat (𝑙𝑘)))))) finSupp 0 )
 
Theorempmatcollpw1lem1 21312* Lemma 1 for pmatcollpw1 21314. (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 21313* Lemma 2 for pmatcollpw1 21314: 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 21314* 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 21315* Lemma for pmatcollpw2 21316. (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 21316* Write a polynomial matrix as a 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 21317 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 21308 (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 21318 Lemma for pmatcollpw 21319. (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 21319* Write a polynomial matrix (over a commutative ring) as a 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 21320* 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, 3-Jul-2022.)
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 21321* Lemma for pmatcollpw3 21322 and pmatcollpw3fi 21323: Write a polynomial matrix (over a commutative ring) as a 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 21322* Write a polynomial matrix (over a commutative ring) as a 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 21323* 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 21324* Lemma 1 for pmatcollpw3fi1 21326. (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 21325* Lemma 2 for pmatcollpw3fi1 21326. (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 21326* 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 21327 Lemma 1 for pmatcollpwscmat 21329. (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 21328 Lemma 2 for pmatcollpwscmat 21329. (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 21329* Write a scalar matrix over polynomials (over a commutative ring) as a 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 21361, 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 21353 and pm2mprngiso 21360), 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 21349).

In this section, the following conventions are mostly used:

  • 𝑅 is a (unital) ring (see df-ring 19230)
  • 𝑃 = (Poly1𝑅) is the polynomial algebra over (the ring) 𝑅 (see df-ply1 20280)
    • 𝐾 = (Base‘𝑃) is its base set (see df-base 16479)
    • 𝑌 = (var1𝑅) is its variable (see df-vr1 20279)
    • · = ( ·𝑠𝑃) is its scalar multiplication (see df-vsca 16572 or lmodvscl 19582)
    • 𝐸 = (.g‘(mulGrp‘𝑃)) is its exponentiation (see df-mulg 18165)
  • 𝐴 = (𝑁 Mat 𝑅) is the algebra of N x N matrices over (the ring) 𝑅 (see df-mat 20947)
  • 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 21330 Extend class notation with the transformation of a polynomial matrix into a polynomial over matrices.
class pMatToMatPoly
 
Definitiondf-pm2mp 21331* 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 21332* Lemma for pm2mpf1 21337. (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 21333* 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 21334* 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 21335 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 21336 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 21337 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 21338 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 21339 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 21340* 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 21341* Lemma for mply1topmatcl 21343. (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 21342* 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 21349). (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 21343* 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 21344* Lemma 1 for mp2pm2mp 21349. (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 21345* Lemma 2 for mp2pm2mp 21349. (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 21346* Lemma 3 for mp2pm2mp 21349. (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 21347* Lemma 4 for mp2pm2mp 21349. (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 21348* Lemma 5 for mp2pm2mp 21349. (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 21349* 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 21350* Lemma 2 for pm2mpghm 21354. (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 21351 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 21352 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 21353 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 21354 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 21355 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 21356* Lemma 1 for pm2mpmhm 21358. (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 21357* Lemma 2 for pm2mpmhm 21358. (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 21358 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 21359 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 21360 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 21361 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 21362 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 21363* 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 21365) 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 21364 Extend class notation with the characteristic polynomial.
class CharPlyMat
 
Definitiondf-chpmat 21365* 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 21366 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 21367 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 21368* 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 21369 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 21370 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 21371* 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 21372 The characteristic polynomial of the empty matrix. (Contributed by AV, 6-Aug-2019.)
𝐶 = (∅ CharPlyMat 𝑅)       (𝑅 ∈ Ring → (𝐶‘∅) = (1r‘(Poly1𝑅)))
 
Theoremchpmat1dlem 21373 Lemma for chpmat1d 21374. (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 21374 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 21375 Lemma 0 for chpdmat 21379. (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 21376 Lemma 1 for chpdmat 21379. (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 21377 Lemma 2 for chpdmat 21379. (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 21378 Lemma 3 for chpdmat 21379. (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 21379* 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 21380* 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 21381* 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 21382* 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 21383* 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 21384 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 21385 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 21386 The characteristic polynomial of a matrix multiplied with the identity matrix is a scalar matrix. (Contributed by AV, 30-Oct-2019.) (Revised by AV, 5-Jul-2022.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐶 = (𝑁 CharPlyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐸 = (Base‘𝑃)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝐾 = (Base‘𝑌)    &    · = ( ·𝑠𝑌)    &    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 21416. 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 21404, cayhamlem3 21425 and cayhamlem4 21426.

 
Theoremfvmptnn04if 21387* 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 21388* 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 21389* 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 21390* 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 21391* 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 21392* 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 21393* 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 21394* 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 21395* 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 21396* 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 21397* 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 21398* 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 21399* 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 21400* 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 )
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