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Type | Label | Description |
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Statement | ||
Definition | df-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))) | ||
Theorem | decpmatval0 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.) |
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐾 ∈ ℕ0) → (𝑀 decompPMat 𝐾) = (𝑖 ∈ dom dom 𝑀, 𝑗 ∈ dom dom 𝑀 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝐾))) | ||
Theorem | decpmatval 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) ⇒ ⊢ ((𝑀 ∈ 𝐵 ∧ 𝐾 ∈ ℕ0) → (𝑀 decompPMat 𝐾) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖𝑀𝑗))‘𝐾))) | ||
Theorem | decpmate 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 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ ℕ0) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝑀 decompPMat 𝐾)𝐽) = ((coe1‘(𝐼𝑀𝐽))‘𝐾)) | ||
Theorem | decpmatcl 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 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐷 = (Base‘𝐴) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵 ∧ 𝐾 ∈ ℕ0) → (𝑀 decompPMat 𝐾) ∈ 𝐷) | ||
Theorem | decpmataa0 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 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 0 = (0g‘𝐴) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → (𝑀 decompPMat 𝑥) = 0 )) | ||
Theorem | decpmatfsupp 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 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 0 = (0g‘𝐴) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ (𝑀 decompPMat 𝑘)) finSupp 0 ) | ||
Theorem | decpmatid 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 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐼 = (1r‘𝐶) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 0 = (0g‘𝐴) & ⊢ 1 = (1r‘𝐴) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (𝐼 decompPMat 𝐾) = if(𝐾 = 0, 1 , 0 )) | ||
Theorem | decpmatmullem 21309* | Lemma for decpmatmul 21310. (Contributed by AV, 20-Oct-2019.) (Revised by AV, 3-Dec-2019.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐾 ∈ ℕ0)) → (𝐼((𝑈(.r‘𝐶)𝑊) decompPMat 𝐾)𝐽) = (𝑅 Σg (𝑡 ∈ 𝑁 ↦ (𝑅 Σg (𝑙 ∈ (0...𝐾) ↦ (((coe1‘(𝐼𝑈𝑡))‘𝑙)(.r‘𝑅)((coe1‘(𝑡𝑊𝐽))‘(𝐾 − 𝑙)))))))) | ||
Theorem | decpmatmul 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 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐴 = (𝑁 Mat 𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ (𝑈 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → ((𝑈(.r‘𝐶)𝑊) decompPMat 𝐾) = (𝐴 Σg (𝑘 ∈ (0...𝐾) ↦ ((𝑈 decompPMat 𝑘)(.r‘𝐴)(𝑊 decompPMat (𝐾 − 𝑘)))))) | ||
Theorem | decpmatmulsumfsupp 21311* | Lemma 0 for pm2mpmhm 21358. (Contributed by AV, 21-Oct-2019.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ · = (.r‘𝐴) & ⊢ 0 = (0g‘𝐴) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑙 ∈ ℕ0 ↦ (𝐴 Σg (𝑘 ∈ (0...𝑙) ↦ ((𝑥 decompPMat 𝑘) · (𝑦 decompPMat (𝑙 − 𝑘)))))) finSupp 0 ) | ||
Theorem | pmatcollpw1lem1 21312* | Lemma 1 for pmatcollpw1 21314. (Contributed by AV, 28-Sep-2019.) (Revised by AV, 3-Dec-2019.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ × = ( ·𝑠 ‘𝑃) & ⊢ ↑ = (.g‘(mulGrp‘𝑃)) & ⊢ 𝑋 = (var1‘𝑅) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → (𝑛 ∈ ℕ0 ↦ ((𝐼(𝑀 decompPMat 𝑛)𝐽) × (𝑛 ↑ 𝑋))) finSupp (0g‘𝑃)) | ||
Theorem | pmatcollpw1lem2 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 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ × = ( ·𝑠 ‘𝑃) & ⊢ ↑ = (.g‘(mulGrp‘𝑃)) & ⊢ 𝑋 = (var1‘𝑅) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (𝑎𝑀𝑏) = (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝑎(𝑀 decompPMat 𝑛)𝑏) × (𝑛 ↑ 𝑋))))) | ||
Theorem | pmatcollpw1 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 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ × = ( ·𝑠 ‘𝑃) & ⊢ ↑ = (.g‘(mulGrp‘𝑃)) & ⊢ 𝑋 = (var1‘𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑀 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑛 ∈ ℕ0 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋)))))) | ||
Theorem | pmatcollpw2lem 21315* | Lemma for pmatcollpw2 21316. (Contributed by AV, 3-Oct-2019.) (Revised by AV, 3-Dec-2019.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ × = ( ·𝑠 ‘𝑃) & ⊢ ↑ = (.g‘(mulGrp‘𝑃)) & ⊢ 𝑋 = (var1‘𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋)))) finSupp (0g‘𝐶)) | ||
Theorem | pmatcollpw2 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 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ × = ( ·𝑠 ‘𝑃) & ⊢ ↑ = (.g‘(mulGrp‘𝑃)) & ⊢ 𝑋 = (var1‘𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝑀 = (𝐶 Σg (𝑛 ∈ ℕ0 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 ↑ 𝑋)))))) | ||
Theorem | monmatcollpw 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 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐾 = (Base‘𝐴) & ⊢ 0 = (0g‘𝐴) & ⊢ ↑ = (.g‘(mulGrp‘𝑃)) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ · = ( ·𝑠 ‘𝐶) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → (((𝐿 ↑ 𝑋) · (𝑇‘𝑀)) decompPMat 𝐼) = if(𝐼 = 𝐿, 𝑀, 0 )) | ||
Theorem | pmatcollpwlem 21318 | Lemma for pmatcollpw 21319. (Contributed by AV, 26-Oct-2019.) (Revised by AV, 4-Dec-2019.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ ∗ = ( ·𝑠 ‘𝐶) & ⊢ ↑ = (.g‘(mulGrp‘𝑃)) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) ⇒ ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → ((𝑎(𝑀 decompPMat 𝑛)𝑏)( ·𝑠 ‘𝑃)(𝑛 ↑ 𝑋)) = (𝑎((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛)))𝑏)) | ||
Theorem | pmatcollpw 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 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ ∗ = ( ·𝑠 ‘𝐶) & ⊢ ↑ = (.g‘(mulGrp‘𝑃)) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑀 = (𝐶 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛)))))) | ||
Theorem | pmatcollpwfi 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 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ ∗ = ( ·𝑠 ‘𝐶) & ⊢ ↑ = (.g‘(mulGrp‘𝑃)) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ0 𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛)))))) | ||
Theorem | pmatcollpw3lem 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 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ ∗ = ( ·𝑠 ‘𝐶) & ⊢ ↑ = (.g‘(mulGrp‘𝑃)) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐷 = (Base‘𝐴) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ⊆ ℕ0 ∧ 𝐼 ≠ ∅)) → (𝑀 = (𝐶 Σg (𝑛 ∈ 𝐼 ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))))) → ∃𝑓 ∈ (𝐷 ↑m 𝐼)𝑀 = (𝐶 Σg (𝑛 ∈ 𝐼 ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛))))))) | ||
Theorem | pmatcollpw3 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 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ ∗ = ( ·𝑠 ‘𝐶) & ⊢ ↑ = (.g‘(mulGrp‘𝑃)) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐷 = (Base‘𝐴) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑓 ∈ (𝐷 ↑m ℕ0)𝑀 = (𝐶 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛)))))) | ||
Theorem | pmatcollpw3fi 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 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ ∗ = ( ·𝑠 ‘𝐶) & ⊢ ↑ = (.g‘(mulGrp‘𝑃)) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐷 = (Base‘𝐴) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ0 ∃𝑓 ∈ (𝐷 ↑m (0...𝑠))𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛)))))) | ||
Theorem | pmatcollpw3fi1lem1 21324* | Lemma 1 for pmatcollpw3fi1 21326. (Contributed by AV, 6-Nov-2019.) (Revised by AV, 4-Dec-2019.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ ∗ = ( ·𝑠 ‘𝐶) & ⊢ ↑ = (.g‘(mulGrp‘𝑃)) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐷 = (Base‘𝐴) & ⊢ 0 = (0g‘𝐴) & ⊢ 𝐻 = (𝑙 ∈ (0...1) ↦ if(𝑙 = 0, (𝐺‘0), 0 )) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐺 ∈ (𝐷 ↑m {0}) ∧ 𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐺‘𝑛)))))) → 𝑀 = (𝐶 Σg (𝑛 ∈ (0...1) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝐻‘𝑛)))))) | ||
Theorem | pmatcollpw3fi1lem2 21325* | Lemma 2 for pmatcollpw3fi1 21326. (Contributed by AV, 6-Nov-2019.) (Revised by AV, 4-Dec-2019.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ ∗ = ( ·𝑠 ‘𝐶) & ⊢ ↑ = (.g‘(mulGrp‘𝑃)) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐷 = (Base‘𝐴) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (∃𝑓 ∈ (𝐷 ↑m {0})𝑀 = (𝐶 Σg (𝑛 ∈ {0} ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛))))) → ∃𝑠 ∈ ℕ ∃𝑓 ∈ (𝐷 ↑m (0...𝑠))𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛))))))) | ||
Theorem | pmatcollpw3fi1 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 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ ∗ = ( ·𝑠 ‘𝐶) & ⊢ ↑ = (.g‘(mulGrp‘𝑃)) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐷 = (Base‘𝐴) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ ∃𝑓 ∈ (𝐷 ↑m (0...𝑠))𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑓‘𝑛)))))) | ||
Theorem | pmatcollpwscmatlem1 21327 | Lemma 1 for pmatcollpwscmat 21329. (Contributed by AV, 2-Nov-2019.) (Revised by AV, 4-Dec-2019.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ ∗ = ( ·𝑠 ‘𝐶) & ⊢ ↑ = (.g‘(mulGrp‘𝑃)) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐷 = (Base‘𝐴) & ⊢ 𝑈 = (algSc‘𝑃) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐸 = (Base‘𝑃) & ⊢ 𝑆 = (algSc‘𝑃) & ⊢ 1 = (1r‘𝐶) & ⊢ 𝑀 = (𝑄 ∗ 1 ) ⇒ ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (((coe1‘(𝑎𝑀𝑏))‘𝐿)( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))) = if(𝑎 = 𝑏, (𝑈‘((coe1‘𝑄)‘𝐿)), (0g‘𝑃))) | ||
Theorem | pmatcollpwscmatlem2 21328 | Lemma 2 for pmatcollpwscmat 21329. (Contributed by AV, 2-Nov-2019.) (Revised by AV, 4-Dec-2019.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ ∗ = ( ·𝑠 ‘𝐶) & ⊢ ↑ = (.g‘(mulGrp‘𝑃)) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐷 = (Base‘𝐴) & ⊢ 𝑈 = (algSc‘𝑃) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐸 = (Base‘𝑃) & ⊢ 𝑆 = (algSc‘𝑃) & ⊢ 1 = (1r‘𝐶) & ⊢ 𝑀 = (𝑄 ∗ 1 ) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐿 ∈ ℕ0 ∧ 𝑄 ∈ 𝐸)) → (𝑇‘(𝑀 decompPMat 𝐿)) = ((𝑈‘((coe1‘𝑄)‘𝐿)) ∗ 1 )) | ||
Theorem | pmatcollpwscmat 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 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ ∗ = ( ·𝑠 ‘𝐶) & ⊢ ↑ = (.g‘(mulGrp‘𝑃)) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐷 = (Base‘𝐴) & ⊢ 𝑈 = (algSc‘𝑃) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐸 = (Base‘𝑃) & ⊢ 𝑆 = (algSc‘𝑃) & ⊢ 1 = (1r‘𝐶) & ⊢ 𝑀 = (𝑄 ∗ 1 ) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑄 ∈ 𝐸) → 𝑀 = (𝐶 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛 ↑ 𝑋) ∗ ((𝑈‘((coe1‘𝑄)‘𝑛)) ∗ 1 ))))) | ||
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:
| ||
Syntax | cpm2mp 21330 | Extend class notation with the transformation of a polynomial matrix into a polynomial over matrices. |
class pMatToMatPoly | ||
Definition | df-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 (Poly1‘𝑟))) ↦ ⦋(𝑛 Mat 𝑟) / 𝑎⦌⦋(Poly1‘𝑎) / 𝑞⦌(𝑞 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠 ‘𝑞)(𝑘(.g‘(mulGrp‘𝑞))(var1‘𝑎))))))) | ||
Theorem | pm2mpf1lem 21332* | Lemma for pm2mpf1 21337. (Contributed by AV, 14-Oct-2019.) (Revised by AV, 4-Dec-2019.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ ∗ = ( ·𝑠 ‘𝑄) & ⊢ ↑ = (.g‘(mulGrp‘𝑄)) & ⊢ 𝑋 = (var1‘𝐴) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑄 = (Poly1‘𝐴) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑈 ∈ 𝐵 ∧ 𝐾 ∈ ℕ0)) → ((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑈 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))))‘𝐾) = (𝑈 decompPMat 𝐾)) | ||
Theorem | pm2mpval 21333* | Value of the transformation of a polynomial matrix into a polynomial over matrices. (Contributed by AV, 5-Dec-2019.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ ∗ = ( ·𝑠 ‘𝑄) & ⊢ ↑ = (.g‘(mulGrp‘𝑄)) & ⊢ 𝑋 = (var1‘𝐴) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑄 = (Poly1‘𝐴) & ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑇 = (𝑚 ∈ 𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))))) | ||
Theorem | pm2mpfval 21334* | A polynomial matrix transformed into a polynomial over matrices. (Contributed by AV, 4-Oct-2019.) (Revised by AV, 5-Dec-2019.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ ∗ = ( ·𝑠 ‘𝑄) & ⊢ ↑ = (.g‘(mulGrp‘𝑄)) & ⊢ 𝑋 = (var1‘𝐴) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑄 = (Poly1‘𝐴) & ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))))) | ||
Theorem | pm2mpcl 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 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ ∗ = ( ·𝑠 ‘𝑄) & ⊢ ↑ = (.g‘(mulGrp‘𝑄)) & ⊢ 𝑋 = (var1‘𝐴) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑄 = (Poly1‘𝐴) & ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅) & ⊢ 𝐿 = (Base‘𝑄) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ 𝐿) | ||
Theorem | pm2mpf 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 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ ∗ = ( ·𝑠 ‘𝑄) & ⊢ ↑ = (.g‘(mulGrp‘𝑄)) & ⊢ 𝑋 = (var1‘𝐴) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑄 = (Poly1‘𝐴) & ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅) & ⊢ 𝐿 = (Base‘𝑄) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵⟶𝐿) | ||
Theorem | pm2mpf1 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 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ ∗ = ( ·𝑠 ‘𝑄) & ⊢ ↑ = (.g‘(mulGrp‘𝑄)) & ⊢ 𝑋 = (var1‘𝐴) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑄 = (Poly1‘𝐴) & ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅) & ⊢ 𝐿 = (Base‘𝑄) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵–1-1→𝐿) | ||
Theorem | pm2mpcoe1 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 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ ∗ = ( ·𝑠 ‘𝑄) & ⊢ ↑ = (.g‘(mulGrp‘𝑄)) & ⊢ 𝑋 = (var1‘𝐴) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑄 = (Poly1‘𝐴) & ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐵 ∧ 𝐾 ∈ ℕ0)) → ((coe1‘(𝑇‘𝑀))‘𝐾) = (𝑀 decompPMat 𝐾)) | ||
Theorem | idpm2idmp 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 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ ∗ = ( ·𝑠 ‘𝑄) & ⊢ ↑ = (.g‘(mulGrp‘𝑄)) & ⊢ 𝑋 = (var1‘𝐴) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑄 = (Poly1‘𝐴) & ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑇‘(1r‘𝐶)) = (1r‘𝑄)) | ||
Theorem | mptcoe1matfsupp 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 = (Poly1‘𝐴) & ⊢ 𝐿 = (Base‘𝑄) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → (𝑘 ∈ ℕ0 ↦ (𝐼((coe1‘𝑂)‘𝑘)𝐽)) finSupp (0g‘𝑅)) | ||
Theorem | mply1topmatcllem 21341* | Lemma for mply1topmatcl 21343. (Contributed by AV, 6-Oct-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑄 = (Poly1‘𝐴) & ⊢ 𝐿 = (Base‘𝑄) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ · = ( ·𝑠 ‘𝑃) & ⊢ 𝐸 = (.g‘(mulGrp‘𝑃)) & ⊢ 𝑌 = (var1‘𝑅) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → (𝑘 ∈ ℕ0 ↦ ((𝐼((coe1‘𝑂)‘𝑘)𝐽) · (𝑘𝐸𝑌))) finSupp (0g‘𝑃)) | ||
Theorem | mply1topmatval 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‘(𝐼‘𝑂)) = 𝑂) (see mp2pm2mp 21349). (Contributed by AV, 6-Oct-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑄 = (Poly1‘𝐴) & ⊢ 𝐿 = (Base‘𝑄) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ · = ( ·𝑠 ‘𝑃) & ⊢ 𝐸 = (.g‘(mulGrp‘𝑃)) & ⊢ 𝑌 = (var1‘𝑅) & ⊢ 𝐼 = (𝑝 ∈ 𝐿 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)))))) ⇒ ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑂 ∈ 𝐿) → (𝐼‘𝑂) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌)))))) | ||
Theorem | mply1topmatcl 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 = (Poly1‘𝐴) & ⊢ 𝐿 = (Base‘𝑄) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ · = ( ·𝑠 ‘𝑃) & ⊢ 𝐸 = (.g‘(mulGrp‘𝑃)) & ⊢ 𝑌 = (var1‘𝑅) & ⊢ 𝐼 = (𝑝 ∈ 𝐿 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)))))) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝐼‘𝑂) ∈ 𝐵) | ||
Theorem | mp2pm2mplem1 21344* | Lemma 1 for mp2pm2mp 21349. (Contributed by AV, 9-Oct-2019.) (Revised by AV, 5-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑄 = (Poly1‘𝐴) & ⊢ 𝐿 = (Base‘𝑄) & ⊢ · = ( ·𝑠 ‘𝑃) & ⊢ 𝐸 = (.g‘(mulGrp‘𝑃)) & ⊢ 𝑌 = (var1‘𝑅) & ⊢ 𝐼 = (𝑝 ∈ 𝐿 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)))))) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝐼‘𝑂) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌)))))) | ||
Theorem | mp2pm2mplem2 21345* | Lemma 2 for mp2pm2mp 21349. (Contributed by AV, 10-Oct-2019.) (Revised by AV, 5-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑄 = (Poly1‘𝐴) & ⊢ 𝐿 = (Base‘𝑄) & ⊢ · = ( ·𝑠 ‘𝑃) & ⊢ 𝐸 = (.g‘(mulGrp‘𝑃)) & ⊢ 𝑌 = (var1‘𝑅) & ⊢ 𝐼 = (𝑝 ∈ 𝐿 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)))))) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌))))) ∈ 𝐵) | ||
Theorem | mp2pm2mplem3 21346* | Lemma 3 for mp2pm2mp 21349. (Contributed by AV, 10-Oct-2019.) (Revised by AV, 5-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑄 = (Poly1‘𝐴) & ⊢ 𝐿 = (Base‘𝑄) & ⊢ · = ( ·𝑠 ‘𝑃) & ⊢ 𝐸 = (.g‘(mulGrp‘𝑃)) & ⊢ 𝑌 = (var1‘𝑅) & ⊢ 𝐼 = (𝑝 ∈ 𝐿 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)))))) & ⊢ 𝑃 = (Poly1‘𝑅) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐾 ∈ ℕ0) → ((𝐼‘𝑂) decompPMat 𝐾) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑂)‘𝑘)𝑗) · (𝑘𝐸𝑌)))))‘𝐾))) | ||
Theorem | mp2pm2mplem4 21347* | Lemma 4 for mp2pm2mp 21349. (Contributed by AV, 12-Oct-2019.) (Revised by AV, 5-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑄 = (Poly1‘𝐴) & ⊢ 𝐿 = (Base‘𝑄) & ⊢ · = ( ·𝑠 ‘𝑃) & ⊢ 𝐸 = (.g‘(mulGrp‘𝑃)) & ⊢ 𝑌 = (var1‘𝑅) & ⊢ 𝐼 = (𝑝 ∈ 𝐿 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)))))) & ⊢ 𝑃 = (Poly1‘𝑅) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) ∧ 𝐾 ∈ ℕ0) → ((𝐼‘𝑂) decompPMat 𝐾) = ((coe1‘𝑂)‘𝐾)) | ||
Theorem | mp2pm2mplem5 21348* | Lemma 5 for mp2pm2mp 21349. (Contributed by AV, 12-Oct-2019.) (Revised by AV, 5-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑄 = (Poly1‘𝐴) & ⊢ 𝐿 = (Base‘𝑄) & ⊢ · = ( ·𝑠 ‘𝑃) & ⊢ 𝐸 = (.g‘(mulGrp‘𝑃)) & ⊢ 𝑌 = (var1‘𝑅) & ⊢ 𝐼 = (𝑝 ∈ 𝐿 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)))))) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ ∗ = ( ·𝑠 ‘𝑄) & ⊢ ↑ = (.g‘(mulGrp‘𝑄)) & ⊢ 𝑋 = (var1‘𝐴) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝑘 ∈ ℕ0 ↦ (((𝐼‘𝑂) decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))) finSupp (0g‘𝑄)) | ||
Theorem | mp2pm2mp 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 = (Poly1‘𝐴) & ⊢ 𝐿 = (Base‘𝑄) & ⊢ · = ( ·𝑠 ‘𝑃) & ⊢ 𝐸 = (.g‘(mulGrp‘𝑃)) & ⊢ 𝑌 = (var1‘𝑅) & ⊢ 𝐼 = (𝑝 ∈ 𝐿 ↦ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑃 Σg (𝑘 ∈ ℕ0 ↦ ((𝑖((coe1‘𝑝)‘𝑘)𝑗) · (𝑘𝐸𝑌)))))) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑂 ∈ 𝐿) → (𝑇‘(𝐼‘𝑂)) = 𝑂) | ||
Theorem | pm2mpghmlem2 21350* | Lemma 2 for pm2mpghm 21354. (Contributed by AV, 15-Oct-2019.) (Revised by AV, 4-Dec-2019.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ ∗ = ( ·𝑠 ‘𝑄) & ⊢ ↑ = (.g‘(mulGrp‘𝑄)) & ⊢ 𝑋 = (var1‘𝐴) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑄 = (Poly1‘𝐴) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑘 ∈ ℕ0 ↦ ((𝑀 decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))) finSupp (0g‘𝑄)) | ||
Theorem | pm2mpghmlem1 21351 | Lemma 1 for pm2mpghm . (Contributed by AV, 15-Oct-2019.) (Revised by AV, 4-Dec-2019.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ ∗ = ( ·𝑠 ‘𝑄) & ⊢ ↑ = (.g‘(mulGrp‘𝑄)) & ⊢ 𝑋 = (var1‘𝐴) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑄 = (Poly1‘𝐴) & ⊢ 𝐿 = (Base‘𝑄) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝐾 ∈ ℕ0) → ((𝑀 decompPMat 𝐾) ∗ (𝐾 ↑ 𝑋)) ∈ 𝐿) | ||
Theorem | pm2mpfo 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 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ ∗ = ( ·𝑠 ‘𝑄) & ⊢ ↑ = (.g‘(mulGrp‘𝑄)) & ⊢ 𝑋 = (var1‘𝐴) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑄 = (Poly1‘𝐴) & ⊢ 𝐿 = (Base‘𝑄) & ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵–onto→𝐿) | ||
Theorem | pm2mpf1o 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 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ ∗ = ( ·𝑠 ‘𝑄) & ⊢ ↑ = (.g‘(mulGrp‘𝑄)) & ⊢ 𝑋 = (var1‘𝐴) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑄 = (Poly1‘𝐴) & ⊢ 𝐿 = (Base‘𝑄) & ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵–1-1-onto→𝐿) | ||
Theorem | pm2mpghm 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 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ ∗ = ( ·𝑠 ‘𝑄) & ⊢ ↑ = (.g‘(mulGrp‘𝑄)) & ⊢ 𝑋 = (var1‘𝐴) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑄 = (Poly1‘𝐴) & ⊢ 𝐿 = (Base‘𝑄) & ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐶 GrpHom 𝑄)) | ||
Theorem | pm2mpgrpiso 21355 | The transformation of polynomial matrices into polynomials over matrices is an additive group isomorphism. (Contributed by AV, 17-Oct-2019.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ ∗ = ( ·𝑠 ‘𝑄) & ⊢ ↑ = (.g‘(mulGrp‘𝑄)) & ⊢ 𝑋 = (var1‘𝐴) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑄 = (Poly1‘𝐴) & ⊢ 𝐿 = (Base‘𝑄) & ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐶 GrpIso 𝑄)) | ||
Theorem | pm2mpmhmlem1 21356* | Lemma 1 for pm2mpmhm 21358. (Contributed by AV, 21-Oct-2019.) (Revised by AV, 6-Dec-2019.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ ∗ = ( ·𝑠 ‘𝑄) & ⊢ ↑ = (.g‘(mulGrp‘𝑄)) & ⊢ 𝑋 = (var1‘𝐴) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑄 = (Poly1‘𝐴) & ⊢ 𝐿 = (Base‘𝑄) & ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑙 ∈ ℕ0 ↦ ((𝐴 Σg (𝑘 ∈ (0...𝑙) ↦ ((𝑥 decompPMat 𝑘)(.r‘𝐴)(𝑦 decompPMat (𝑙 − 𝑘))))) ∗ (𝑙 ↑ 𝑋))) finSupp (0g‘𝑄)) | ||
Theorem | pm2mpmhmlem2 21357* | Lemma 2 for pm2mpmhm 21358. (Contributed by AV, 22-Oct-2019.) (Revised by AV, 6-Dec-2019.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑄 = (Poly1‘𝐴) & ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅) & ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑇‘(𝑥(.r‘𝐶)𝑦)) = ((𝑇‘𝑥)(.r‘𝑄)(𝑇‘𝑦))) | ||
Theorem | pm2mpmhm 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 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑄 = (Poly1‘𝐴) & ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ ((mulGrp‘𝐶) MndHom (mulGrp‘𝑄))) | ||
Theorem | pm2mprhm 21359 | The transformation of polynomial matrices into polynomials over matrices is a ring homomorphism. (Contributed by AV, 22-Oct-2019.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑄 = (Poly1‘𝐴) & ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐶 RingHom 𝑄)) | ||
Theorem | pm2mprngiso 21360 | The transformation of polynomial matrices into polynomials over matrices is a ring isomorphism. (Contributed by AV, 22-Oct-2019.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑄 = (Poly1‘𝐴) & ⊢ 𝑇 = (𝑁 pMatToMatPoly 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐶 RingIso 𝑄)) | ||
Theorem | pmmpric 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 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑄 = (Poly1‘𝐴) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ≃𝑟 𝑄) | ||
Theorem | monmat2matmon 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 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ ∗ = ( ·𝑠 ‘𝑄) & ⊢ ↑ = (.g‘(mulGrp‘𝑄)) & ⊢ 𝑋 = (var1‘𝐴) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐾 = (Base‘𝐴) & ⊢ 𝑄 = (Poly1‘𝐴) & ⊢ 𝐼 = (𝑁 pMatToMatPoly 𝑅) & ⊢ 𝐸 = (.g‘(mulGrp‘𝑃)) & ⊢ 𝑌 = (var1‘𝑅) & ⊢ · = ( ·𝑠 ‘𝐶) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) → (𝐼‘((𝐿𝐸𝑌) · (𝑇‘𝑀))) = (𝑀 ∗ (𝐿 ↑ 𝑋))) | ||
Theorem | pm2mp 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 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ ∗ = ( ·𝑠 ‘𝑄) & ⊢ ↑ = (.g‘(mulGrp‘𝑄)) & ⊢ 𝑋 = (var1‘𝐴) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐾 = (Base‘𝐴) & ⊢ 𝑄 = (Poly1‘𝐴) & ⊢ 𝐼 = (𝑁 pMatToMatPoly 𝑅) & ⊢ 𝐸 = (.g‘(mulGrp‘𝑃)) & ⊢ 𝑌 = (var1‘𝑅) & ⊢ · = ( ·𝑠 ‘𝐶) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) → (𝐼‘(𝐶 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛)))))) = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ ((𝑀‘𝑛) ∗ (𝑛 ↑ 𝑋))))) | ||
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. | ||
The characteristic polynomial of a matrix 𝐴 is the determinat of the characteristic matrix of 𝐴: (𝑡𝐼 − 𝐴). | ||
Syntax | cchpmat 21364 | Extend class notation with the characteristic polynomial. |
class CharPlyMat | ||
Definition | df-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 (Poly1‘𝑟))‘(((var1‘𝑟)( ·𝑠 ‘(𝑛 Mat (Poly1‘𝑟)))(1r‘(𝑛 Mat (Poly1‘𝑟))))(-g‘(𝑛 Mat (Poly1‘𝑟)))((𝑛 matToPolyMat 𝑟)‘𝑚))))) | ||
Theorem | chmatcl 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 = (Poly1‘𝑅) & ⊢ 𝑌 = (𝑁 Mat 𝑃) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ − = (-g‘𝑌) & ⊢ · = ( ·𝑠 ‘𝑌) & ⊢ 1 = (1r‘𝑌) & ⊢ 𝐻 = ((𝑋 · 1 ) − (𝑇‘𝑀)) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → 𝐻 ∈ (Base‘𝑌)) | ||
Theorem | chmatval 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 = (Poly1‘𝑅) & ⊢ 𝑌 = (𝑁 Mat 𝑃) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ − = (-g‘𝑌) & ⊢ · = ( ·𝑠 ‘𝑌) & ⊢ 1 = (1r‘𝑌) & ⊢ 𝐻 = ((𝑋 · 1 ) − (𝑇‘𝑀)) & ⊢ ∼ = (-g‘𝑃) & ⊢ 0 = (0g‘𝑃) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼𝐻𝐽) = if(𝐼 = 𝐽, (𝑋 ∼ (𝐼(𝑇‘𝑀)𝐽)), ( 0 ∼ (𝐼(𝑇‘𝑀)𝐽)))) | ||
Theorem | chpmatfval 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 = (Poly1‘𝑅) & ⊢ 𝑌 = (𝑁 Mat 𝑃) & ⊢ 𝐷 = (𝑁 maDet 𝑃) & ⊢ − = (-g‘𝑌) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ · = ( ·𝑠 ‘𝑌) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 1 = (1r‘𝑌) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐶 = (𝑚 ∈ 𝐵 ↦ (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑚))))) | ||
Theorem | chpmatval 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 = (Poly1‘𝑅) & ⊢ 𝑌 = (𝑁 Mat 𝑃) & ⊢ 𝐷 = (𝑁 maDet 𝑃) & ⊢ − = (-g‘𝑌) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ · = ( ·𝑠 ‘𝑌) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 1 = (1r‘𝑌) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) = (𝐷‘((𝑋 · 1 ) − (𝑇‘𝑀)))) | ||
Theorem | chpmatply1 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 = (Poly1‘𝑅) & ⊢ 𝐸 = (Base‘𝑃) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) ∈ 𝐸) | ||
Theorem | chpmatval2 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 = (Poly1‘𝑅) & ⊢ 𝑌 = (𝑁 Mat 𝑃) & ⊢ − = (-g‘𝑌) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ · = ( ·𝑠 ‘𝑌) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 1 = (1r‘𝑌) & ⊢ 𝐺 = (SymGrp‘𝑁) & ⊢ 𝐻 = (Base‘𝐺) & ⊢ 𝑍 = (ℤRHom‘𝑃) & ⊢ 𝑆 = (pmSgn‘𝑁) & ⊢ 𝑈 = (mulGrp‘𝑃) & ⊢ × = (.r‘𝑃) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) = (𝑃 Σg (𝑝 ∈ 𝐻 ↦ (((𝑍 ∘ 𝑆)‘𝑝) × (𝑈 Σg (𝑥 ∈ 𝑁 ↦ ((𝑝‘𝑥)((𝑋 · 1 ) − (𝑇‘𝑀))𝑥))))))) | ||
Theorem | chpmat0d 21372 | The characteristic polynomial of the empty matrix. (Contributed by AV, 6-Aug-2019.) |
⊢ 𝐶 = (∅ CharPlyMat 𝑅) ⇒ ⊢ (𝑅 ∈ Ring → (𝐶‘∅) = (1r‘(Poly1‘𝑅))) | ||
Theorem | chpmat1dlem 21373 | Lemma for chpmat1d 21374. (Contributed by AV, 7-Aug-2019.) |
⊢ 𝐶 = (𝑁 CharPlyMat 𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ − = (-g‘𝑃) & ⊢ 𝑆 = (algSc‘𝑃) & ⊢ 𝐺 = (𝑁 Mat 𝑃) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ (𝑁 = {𝐼} ∧ 𝐼 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → (𝐼((𝑋( ·𝑠 ‘𝐺)(1r‘𝐺))(-g‘𝐺)(𝑇‘𝑀))𝐼) = (𝑋 − (𝑆‘(𝐼𝑀𝐼)))) | ||
Theorem | chpmat1d 21374 | The characteristic polynomial of a matrix with dimension 1. (Contributed by AV, 7-Aug-2019.) |
⊢ 𝐶 = (𝑁 CharPlyMat 𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ − = (-g‘𝑃) & ⊢ 𝑆 = (algSc‘𝑃) ⇒ ⊢ ((𝑅 ∈ CRing ∧ (𝑁 = {𝐼} ∧ 𝐼 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → (𝐶‘𝑀) = (𝑋 − (𝑆‘(𝐼𝑀𝐼)))) | ||
Theorem | chpdmatlem0 21375 | Lemma 0 for chpdmat 21379. (Contributed by AV, 18-Aug-2019.) |
⊢ 𝐶 = (𝑁 CharPlyMat 𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑆 = (algSc‘𝑃) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐺 = (mulGrp‘𝑃) & ⊢ − = (-g‘𝑃) & ⊢ 𝑄 = (𝑁 Mat 𝑃) & ⊢ 1 = (1r‘𝑄) & ⊢ · = ( ·𝑠 ‘𝑄) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑋 · 1 ) ∈ (Base‘𝑄)) | ||
Theorem | chpdmatlem1 21376 | Lemma 1 for chpdmat 21379. (Contributed by AV, 18-Aug-2019.) |
⊢ 𝐶 = (𝑁 CharPlyMat 𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑆 = (algSc‘𝑃) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐺 = (mulGrp‘𝑃) & ⊢ − = (-g‘𝑃) & ⊢ 𝑄 = (𝑁 Mat 𝑃) & ⊢ 1 = (1r‘𝑄) & ⊢ · = ( ·𝑠 ‘𝑄) & ⊢ 𝑍 = (-g‘𝑄) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ((𝑋 · 1 )𝑍(𝑇‘𝑀)) ∈ (Base‘𝑄)) | ||
Theorem | chpdmatlem2 21377 | Lemma 2 for chpdmat 21379. (Contributed by AV, 18-Aug-2019.) |
⊢ 𝐶 = (𝑁 CharPlyMat 𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑆 = (algSc‘𝑃) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐺 = (mulGrp‘𝑃) & ⊢ − = (-g‘𝑃) & ⊢ 𝑄 = (𝑁 Mat 𝑃) & ⊢ 1 = (1r‘𝑄) & ⊢ · = ( ·𝑠 ‘𝑄) & ⊢ 𝑍 = (-g‘𝑄) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) ⇒ ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) ∧ 𝑖 ≠ 𝑗) ∧ (𝑖𝑀𝑗) = 0 ) → (𝑖((𝑋 · 1 )𝑍(𝑇‘𝑀))𝑗) = (0g‘𝑃)) | ||
Theorem | chpdmatlem3 21378 | Lemma 3 for chpdmat 21379. (Contributed by AV, 18-Aug-2019.) |
⊢ 𝐶 = (𝑁 CharPlyMat 𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑆 = (algSc‘𝑃) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐺 = (mulGrp‘𝑃) & ⊢ − = (-g‘𝑃) & ⊢ 𝑄 = (𝑁 Mat 𝑃) & ⊢ 1 = (1r‘𝑄) & ⊢ · = ( ·𝑠 ‘𝑄) & ⊢ 𝑍 = (-g‘𝑄) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ 𝐾 ∈ 𝑁) → (𝐾((𝑋 · 1 )𝑍(𝑇‘𝑀))𝐾) = (𝑋 − (𝑆‘(𝐾𝑀𝐾)))) | ||
Theorem | chpdmat 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 = (Poly1‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑆 = (algSc‘𝑃) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐺 = (mulGrp‘𝑃) & ⊢ − = (-g‘𝑃) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑀𝑗) = 0 )) → (𝐶‘𝑀) = (𝐺 Σg (𝑘 ∈ 𝑁 ↦ (𝑋 − (𝑆‘(𝑘𝑀𝑘)))))) | ||
Theorem | chpscmat 21380* | The characteristic polynomial of a (nonempty!) scalar matrix. (Contributed by AV, 21-Aug-2019.) |
⊢ 𝐶 = (𝑁 CharPlyMat 𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ 𝐺 = (mulGrp‘𝑃) & ⊢ ↑ = (.g‘𝐺) & ⊢ 𝐷 = {𝑚 ∈ (Base‘𝐴) ∣ ∃𝑐 ∈ (Base‘𝑅)∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅))} & ⊢ 𝑆 = (algSc‘𝑃) & ⊢ − = (-g‘𝑃) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = 𝐸)) → (𝐶‘𝑀) = ((♯‘𝑁) ↑ (𝑋 − (𝑆‘𝐸)))) | ||
Theorem | chpscmat0 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 = (Poly1‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ 𝐺 = (mulGrp‘𝑃) & ⊢ ↑ = (.g‘𝐺) & ⊢ 𝐷 = {𝑚 ∈ (Base‘𝐴) ∣ ∃𝑐 ∈ (Base‘𝑅)∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅))} & ⊢ 𝑆 = (algSc‘𝑃) & ⊢ − = (-g‘𝑃) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐼 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = (𝐼𝑀𝐼))) → (𝐶‘𝑀) = ((♯‘𝑁) ↑ (𝑋 − (𝑆‘(𝐼𝑀𝐼))))) | ||
Theorem | chpscmatgsumbin 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 = (Poly1‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ 𝐺 = (mulGrp‘𝑃) & ⊢ ↑ = (.g‘𝐺) & ⊢ 𝐷 = {𝑚 ∈ (Base‘𝐴) ∣ ∃𝑐 ∈ (Base‘𝑅)∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅))} & ⊢ 𝑆 = (algSc‘𝑃) & ⊢ − = (-g‘𝑃) & ⊢ 𝐹 = (.g‘𝑃) & ⊢ 𝐻 = (mulGrp‘𝑅) & ⊢ 𝐸 = (.g‘𝐻) & ⊢ 𝐼 = (invg‘𝑅) & ⊢ · = ( ·𝑠 ‘𝑃) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐽 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = (𝐽𝑀𝐽))) → (𝐶‘𝑀) = (𝑃 Σg (𝑙 ∈ (0...(♯‘𝑁)) ↦ (((♯‘𝑁)C𝑙)𝐹((((♯‘𝑁) − 𝑙)𝐸(𝐼‘(𝐽𝑀𝐽))) · (𝑙 ↑ 𝑋)))))) | ||
Theorem | chpscmatgsummon 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 = (Poly1‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ 𝐺 = (mulGrp‘𝑃) & ⊢ ↑ = (.g‘𝐺) & ⊢ 𝐷 = {𝑚 ∈ (Base‘𝐴) ∣ ∃𝑐 ∈ (Base‘𝑅)∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, (0g‘𝑅))} & ⊢ 𝑆 = (algSc‘𝑃) & ⊢ − = (-g‘𝑃) & ⊢ 𝐹 = (.g‘𝑃) & ⊢ 𝐻 = (mulGrp‘𝑅) & ⊢ 𝐸 = (.g‘𝐻) & ⊢ 𝐼 = (invg‘𝑅) & ⊢ · = ( ·𝑠 ‘𝑃) & ⊢ 𝑍 = (.g‘𝑅) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐷 ∧ 𝐽 ∈ 𝑁 ∧ ∀𝑛 ∈ 𝑁 (𝑛𝑀𝑛) = (𝐽𝑀𝐽))) → (𝐶‘𝑀) = (𝑃 Σg (𝑙 ∈ (0...(♯‘𝑁)) ↦ ((((♯‘𝑁)C𝑙)𝑍(((♯‘𝑁) − 𝑙)𝐸(𝐼‘(𝐽𝑀𝐽)))) · (𝑙 ↑ 𝑋))))) | ||
Theorem | chp0mat 21384 | The characteristic polynomial of the zero matrix. (Contributed by AV, 18-Aug-2019.) |
⊢ 𝐶 = (𝑁 CharPlyMat 𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ 𝐺 = (mulGrp‘𝑃) & ⊢ ↑ = (.g‘𝐺) & ⊢ 0 = (0g‘𝐴) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝐶‘ 0 ) = ((♯‘𝑁) ↑ 𝑋)) | ||
Theorem | chpidmat 21385 | The characteristic polynomial of the identity matrix. (Contributed by AV, 19-Aug-2019.) |
⊢ 𝐶 = (𝑁 CharPlyMat 𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ 𝐺 = (mulGrp‘𝑃) & ⊢ ↑ = (.g‘𝐺) & ⊢ 𝐼 = (1r‘𝐴) & ⊢ 𝑆 = (algSc‘𝑃) & ⊢ 1 = (1r‘𝑅) & ⊢ − = (-g‘𝑃) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝐶‘𝐼) = ((♯‘𝑁) ↑ (𝑋 − (𝑆‘ 1 )))) | ||
Theorem | chmaidscmat 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 = (Poly1‘𝑅) & ⊢ 𝐸 = (Base‘𝑃) & ⊢ 𝑌 = (𝑁 Mat 𝑃) & ⊢ 𝐾 = (Base‘𝑌) & ⊢ · = ( ·𝑠 ‘𝑌) & ⊢ 1 = (1r‘𝑌) & ⊢ 𝑆 = (𝑁 ScMat 𝑃) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝐶‘𝑀) · 1 ) ∈ 𝑆) | ||
In this subsection the function 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) is discussed. This function is involved in the representation of the product of the characteristic matrix of a given matrix and its adjunct as an infinite sum, see cpmadugsum 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. | ||
Theorem | fvmptnn04if 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)))) & ⊢ (𝜑 → 𝑆 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑁 = 0) → 𝑌 = ⦋𝑁 / 𝑛⦌𝐴) & ⊢ ((𝜑 ∧ 0 < 𝑁 ∧ 𝑁 < 𝑆) → 𝑌 = ⦋𝑁 / 𝑛⦌𝐵) & ⊢ ((𝜑 ∧ 𝑁 = 𝑆) → 𝑌 = ⦋𝑁 / 𝑛⦌𝐶) & ⊢ ((𝜑 ∧ 𝑆 < 𝑁) → 𝑌 = ⦋𝑁 / 𝑛⦌𝐷) ⇒ ⊢ (𝜑 → (𝐺‘𝑁) = 𝑌) | ||
Theorem | fvmptnn04ifa 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)))) & ⊢ (𝜑 → 𝑆 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ ((𝜑 ∧ 𝑁 = 0 ∧ ⦋𝑁 / 𝑛⦌𝐴 ∈ 𝑉) → (𝐺‘𝑁) = ⦋𝑁 / 𝑛⦌𝐴) | ||
Theorem | fvmptnn04ifb 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)))) & ⊢ (𝜑 → 𝑆 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ ((𝜑 ∧ (0 < 𝑁 ∧ 𝑁 < 𝑆) ∧ ⦋𝑁 / 𝑛⦌𝐵 ∈ 𝑉) → (𝐺‘𝑁) = ⦋𝑁 / 𝑛⦌𝐵) | ||
Theorem | fvmptnn04ifc 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)))) & ⊢ (𝜑 → 𝑆 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ ((𝜑 ∧ 𝑁 = 𝑆 ∧ ⦋𝑁 / 𝑛⦌𝐶 ∈ 𝑉) → (𝐺‘𝑁) = ⦋𝑁 / 𝑛⦌𝐶) | ||
Theorem | fvmptnn04ifd 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)))) & ⊢ (𝜑 → 𝑆 ∈ ℕ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ ((𝜑 ∧ 𝑆 < 𝑁 ∧ ⦋𝑁 / 𝑛⦌𝐷 ∈ 𝑉) → (𝐺‘𝑁) = ⦋𝑁 / 𝑛⦌𝐷) | ||
Theorem | chfacfisf 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 = (Poly1‘𝑅) & ⊢ 𝑌 = (𝑁 Mat 𝑃) & ⊢ × = (.r‘𝑌) & ⊢ − = (-g‘𝑌) & ⊢ 0 = (0g‘𝑌) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝐺:ℕ0⟶(Base‘𝑌)) | ||
Theorem | chfacfisfcpmat 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 = (Poly1‘𝑅) & ⊢ 𝑌 = (𝑁 Mat 𝑃) & ⊢ × = (.r‘𝑌) & ⊢ − = (-g‘𝑌) & ⊢ 0 = (0g‘𝑌) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) & ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝐺:ℕ0⟶𝑆) | ||
Theorem | chfacffsupp 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 = (Poly1‘𝑅) & ⊢ 𝑌 = (𝑁 Mat 𝑃) & ⊢ × = (.r‘𝑌) & ⊢ − = (-g‘𝑌) & ⊢ 0 = (0g‘𝑌) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → 𝐺 finSupp (0g‘𝑌)) | ||
Theorem | chfacfscmulcl 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 = (Poly1‘𝑅) & ⊢ 𝑌 = (𝑁 Mat 𝑃) & ⊢ × = (.r‘𝑌) & ⊢ − = (-g‘𝑌) & ⊢ 0 = (0g‘𝑌) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ · = ( ·𝑠 ‘𝑌) & ⊢ ↑ = (.g‘(mulGrp‘𝑃)) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝐾 ∈ ℕ0) → ((𝐾 ↑ 𝑋) · (𝐺‘𝐾)) ∈ (Base‘𝑌)) | ||
Theorem | chfacfscmul0 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 = (Poly1‘𝑅) & ⊢ 𝑌 = (𝑁 Mat 𝑃) & ⊢ × = (.r‘𝑌) & ⊢ − = (-g‘𝑌) & ⊢ 0 = (0g‘𝑌) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ · = ( ·𝑠 ‘𝑌) & ⊢ ↑ = (.g‘(mulGrp‘𝑃)) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝐾 ∈ (ℤ≥‘(𝑠 + 2))) → ((𝐾 ↑ 𝑋) · (𝐺‘𝐾)) = 0 ) | ||
Theorem | chfacfscmulfsupp 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 = (Poly1‘𝑅) & ⊢ 𝑌 = (𝑁 Mat 𝑃) & ⊢ × = (.r‘𝑌) & ⊢ − = (-g‘𝑌) & ⊢ 0 = (0g‘𝑌) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ · = ( ·𝑠 ‘𝑌) & ⊢ ↑ = (.g‘(mulGrp‘𝑃)) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑖 ∈ ℕ0 ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖))) finSupp 0 ) | ||
Theorem | chfacfscmulgsum 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 = (Poly1‘𝑅) & ⊢ 𝑌 = (𝑁 Mat 𝑃) & ⊢ × = (.r‘𝑌) & ⊢ − = (-g‘𝑌) & ⊢ 0 = (0g‘𝑌) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ · = ( ·𝑠 ‘𝑌) & ⊢ ↑ = (.g‘(mulGrp‘𝑃)) & ⊢ + = (+g‘𝑌) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ ℕ0 ↦ ((𝑖 ↑ 𝑋) · (𝐺‘𝑖)))) = ((𝑌 Σg (𝑖 ∈ (1...𝑠) ↦ ((𝑖 ↑ 𝑋) · ((𝑇‘(𝑏‘(𝑖 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑖))))))) + ((((𝑠 + 1) ↑ 𝑋) · (𝑇‘(𝑏‘𝑠))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))))) | ||
Theorem | chfacfpmmulcl 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 = (Poly1‘𝑅) & ⊢ 𝑌 = (𝑁 Mat 𝑃) & ⊢ × = (.r‘𝑌) & ⊢ − = (-g‘𝑌) & ⊢ 0 = (0g‘𝑌) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) & ⊢ ↑ = (.g‘(mulGrp‘𝑌)) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝐾 ∈ ℕ0) → ((𝐾 ↑ (𝑇‘𝑀)) × (𝐺‘𝐾)) ∈ (Base‘𝑌)) | ||
Theorem | chfacfpmmul0 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 = (Poly1‘𝑅) & ⊢ 𝑌 = (𝑁 Mat 𝑃) & ⊢ × = (.r‘𝑌) & ⊢ − = (-g‘𝑌) & ⊢ 0 = (0g‘𝑌) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ( 0 − ((𝑇‘𝑀) × (𝑇‘(𝑏‘0)))), if(𝑛 = (𝑠 + 1), (𝑇‘(𝑏‘𝑠)), if((𝑠 + 1) < 𝑛, 0 , ((𝑇‘(𝑏‘(𝑛 − 1))) − ((𝑇‘𝑀) × (𝑇‘(𝑏‘𝑛)))))))) & ⊢ ↑ = (.g‘(mulGrp‘𝑌)) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠))) ∧ 𝐾 ∈ (ℤ≥‘(𝑠 + 2))) → ((𝐾 ↑ (𝑇‘𝑀)) × (𝐺‘𝐾)) = 0 ) |
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