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Type | Label | Description |
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Statement | ||
A polynomial matrix or matrix of polynomials is a matrix whose elements are univariate (or multivariate) polynomials. See Wikipedia "Polynomial matrix" https://en.wikipedia.org/wiki/Polynomial_matrix (18-Nov-2019). In this section, only square matrices whose elements are univariate polynomials are considered. Usually, the ring of such matrices, the ring of n x n matrices over the polynomial ring over a ring 𝑅, is denoted by M(n, R[t]). The elements of this ring are called "polynomial matrices (over the ring 𝑅)" in the following. In Metamath notation, this ring is defined by (𝑁 Mat (Poly1‘𝑅)), usually represented by the class variable 𝐶 (or 𝑌, if 𝐶 is already occupied): 𝐶 = (𝑁 Mat 𝑃) with 𝑃 = (Poly1‘𝑅). | ||
Theorem | pmatring 21301 | The set of polynomial matrices over a ring is a ring. (Contributed by AV, 6-Nov-2019.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring) | ||
Theorem | pmatlmod 21302 | The set of polynomial matrices over a ring is a left module. (Contributed by AV, 6-Nov-2019.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ LMod) | ||
Theorem | pmat0op 21303* | The zero polynomial matrix over a ring represented as operation. (Contributed by AV, 16-Nov-2019.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 0 = (0g‘𝑃) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g‘𝐶) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ 0 )) | ||
Theorem | pmat1op 21304* | The identity polynomial matrix over a ring represented as operation. (Contributed by AV, 16-Nov-2019.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 0 = (0g‘𝑃) & ⊢ 1 = (1r‘𝑃) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r‘𝐶) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, 1 , 0 ))) | ||
Theorem | pmat1ovd 21305 | Entries of the identity polynomial matrix over a ring, deduction form. (Contributed by AV, 16-Nov-2019.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 0 = (0g‘𝑃) & ⊢ 1 = (1r‘𝑃) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ 𝑁) & ⊢ (𝜑 → 𝐽 ∈ 𝑁) & ⊢ 𝑈 = (1r‘𝐶) ⇒ ⊢ (𝜑 → (𝐼𝑈𝐽) = if(𝐼 = 𝐽, 1 , 0 )) | ||
Theorem | pmat0opsc 21306* | The zero polynomial matrix over a ring represented as operation with "lifted scalars" (i.e. elements of the ring underlying the polynomial ring embedded into the polynomial ring by the scalar injection/algebraic scalars function algSc). (Contributed by AV, 16-Nov-2019.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐴 = (algSc‘𝑃) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g‘𝐶) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝐴‘ 0 ))) | ||
Theorem | pmat1opsc 21307* | The identity polynomial matrix over a ring represented as operation with "lifted scalars". (Contributed by AV, 16-Nov-2019.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐴 = (algSc‘𝑃) & ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r‘𝐶) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (𝐴‘ 1 ), (𝐴‘ 0 )))) | ||
Theorem | pmat1ovscd 21308 | Entries of the identity polynomial matrix over a ring represented with "lifted scalars", deduction form. (Contributed by AV, 16-Nov-2019.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐴 = (algSc‘𝑃) & ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ (𝜑 → 𝑁 ∈ Fin) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ 𝑁) & ⊢ (𝜑 → 𝐽 ∈ 𝑁) & ⊢ 𝑈 = (1r‘𝐶) ⇒ ⊢ (𝜑 → (𝐼𝑈𝐽) = if(𝐼 = 𝐽, (𝐴‘ 1 ), (𝐴‘ 0 ))) | ||
Theorem | pmatcoe1fsupp 21309* | For a polynomial matrix there is an upper bound for the coefficients of all the polynomials being not 0. (Contributed by AV, 3-Oct-2019.) (Proof shortened by AV, 28-Nov-2019.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → ∃𝑠 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑠 < 𝑥 → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ((coe1‘(𝑖𝑀𝑗))‘𝑥) = 0 )) | ||
Theorem | 1pmatscmul 21310 | The scalar product of the identity polynomial matrix with a polynomial is a polynomial matrix. (Contributed by AV, 2-Nov-2019.) (Revised by AV, 4-Dec-2019.) |
⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐸 = (Base‘𝑃) & ⊢ ∗ = ( ·𝑠 ‘𝐶) & ⊢ 1 = (1r‘𝐶) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑄 ∈ 𝐸) → (𝑄 ∗ 1 ) ∈ 𝐵) | ||
A constant polynomial matrix is a polynomial matrix whose elements are constant polynomials, i.e. polynomials with no indeterminates. Constant polynomials are obtained by "lifting" a "scalar" (i.e. an element of the underlying ring) into the polynomial ring/algebra by a "scalar injection", i.e. applying the "algebra scalar injection function" algSc (see df-ascl 20087) to a scalar 𝐴 ∈ 𝑅: ((algSc‘𝑃)‘𝐴). In an analogous way, constant polynomial matrices (over the ring 𝑅) are obtained by "lifting" matrices over the ring 𝑅 by the function matToPolyMat (see df-mat2pmat 21315), called "matrix transformation" in the following. In this section it is shown that the set 𝑆 = (𝑁 ConstPolyMat 𝑅) of constant polynomial 𝑁 x 𝑁 matrices over the ring 𝑅 is a subring of the ring of polynomial 𝑁 x 𝑁 matrices over the ring 𝑅 (cpmatsrgpmat 21329) and that 𝑇 = (𝑁 matToPolyMat 𝑅) is a ring isomorphism between the ring of matrices over a ring 𝑅 and the ring of constant polynomial matrices over the ring 𝑅 (see m2cpmrngiso 21366). By this, it is shown that the ring of matrices over a commutative ring is isomorphic to the ring of scalar matrices over the same ring, see matcpmric 21367. Finally 𝐼 = (𝑁 cPolyMatToMat 𝑅), the transformation of a constant polynomial matrix into a matrix, is the inverse function of the matrix transformation 𝑇 = (𝑁 matToPolyMat 𝑅), see m2cpminv 21368. | ||
Syntax | ccpmat 21311 | Extend class notation with the set of all constant polynomial matrices. |
class ConstPolyMat | ||
Syntax | cmat2pmat 21312 | Extend class notation with the transformation of a matrix into a matrix of polynomials. |
class matToPolyMat | ||
Syntax | ccpmat2mat 21313 | Extend class notation with the transformation of a constant polynomial matrix into a matrix. |
class cPolyMatToMat | ||
Definition | df-cpmat 21314* | The set of all constant polynomial matrices, which are all matrices whose entries are constant polynomials (or "scalar polynomials", see ply1sclf 20453). (Contributed by AV, 15-Nov-2019.) |
⊢ ConstPolyMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ {𝑚 ∈ (Base‘(𝑛 Mat (Poly1‘𝑟))) ∣ ∀𝑖 ∈ 𝑛 ∀𝑗 ∈ 𝑛 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g‘𝑟)}) | ||
Definition | df-mat2pmat 21315* | Transformation of a matrix (over a ring) into a matrix over the corresponding polynomial ring. (Contributed by AV, 31-Jul-2019.) |
⊢ matToPolyMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((algSc‘(Poly1‘𝑟))‘(𝑥𝑚𝑦))))) | ||
Definition | df-cpmat2mat 21316* | Transformation of a constant polynomial matrix (over a ring) into a matrix over the corresponding ring. Since this function is the inverse function of matToPolyMat, see m2cpminv 21368, it is also called "inverse matrix transformation" in the following. (Contributed by AV, 14-Dec-2019.) |
⊢ cPolyMatToMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (𝑛 ConstPolyMat 𝑟) ↦ (𝑥 ∈ 𝑛, 𝑦 ∈ 𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)))) | ||
Theorem | cpmat 21317* | Value of the constructor of the set of all constant polynomial matrices, i.e. the set of all 𝑁 x 𝑁 matrices of polynomials over a ring 𝑅. (Contributed by AV, 15-Nov-2019.) |
⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑆 = {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g‘𝑅)}) | ||
Theorem | cpmatpmat 21318 | A constant polynomial matrix is a polynomial matrix. (Contributed by AV, 16-Nov-2019.) |
⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝑆) → 𝑀 ∈ 𝐵) | ||
Theorem | cpmatel 21319* | Property of a constant polynomial matrix. (Contributed by AV, 15-Nov-2019.) |
⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝑀 ∈ 𝑆 ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g‘𝑅))) | ||
Theorem | cpmatelimp 21320* | Implication of a set being a constant polynomial matrix. (Contributed by AV, 18-Nov-2019.) |
⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑀 ∈ 𝑆 → (𝑀 ∈ 𝐵 ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g‘𝑅)))) | ||
Theorem | cpmatel2 21321* | Another property of a constant polynomial matrix. (Contributed by AV, 16-Nov-2019.) (Proof shortened by AV, 27-Nov-2019.) |
⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = (algSc‘𝑃) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑀 ∈ 𝑆 ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑘 ∈ 𝐾 (𝑖𝑀𝑗) = (𝐴‘𝑘))) | ||
Theorem | cpmatelimp2 21322* | Another implication of a set being a constant polynomial matrix. (Contributed by AV, 17-Nov-2019.) |
⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝐴 = (algSc‘𝑃) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑀 ∈ 𝑆 → (𝑀 ∈ 𝐵 ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∃𝑘 ∈ 𝐾 (𝑖𝑀𝑗) = (𝐴‘𝑘)))) | ||
Theorem | 1elcpmat 21323 | The identity of the ring of all polynomial matrices over the ring 𝑅 is a constant polynomial matrix. (Contributed by AV, 16-Nov-2019.) |
⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r‘𝐶) ∈ 𝑆) | ||
Theorem | cpmatacl 21324* | The set of all constant polynomial matrices over a ring 𝑅 is closed under addition. (Contributed by AV, 17-Nov-2019.) (Proof shortened by AV, 28-Nov-2019.) |
⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝐶)𝑦) ∈ 𝑆) | ||
Theorem | cpmatinvcl 21325* | The set of all constant polynomial matrices over a ring 𝑅 is closed under inversion. (Contributed by AV, 17-Nov-2019.) (Proof shortened by AV, 28-Nov-2019.) |
⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥 ∈ 𝑆 ((invg‘𝐶)‘𝑥) ∈ 𝑆) | ||
Theorem | cpmatmcllem 21326* | Lemma for cpmatmcl 21327. (Contributed by AV, 18-Nov-2019.) |
⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 ∀𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘 ∈ 𝑁 ↦ ((𝑖𝑥𝑘)(.r‘𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g‘𝑅)) | ||
Theorem | cpmatmcl 21327* | The set of all constant polynomial matrices over a ring 𝑅 is closed under multiplication. (Contributed by AV, 18-Nov-2019.) |
⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(.r‘𝐶)𝑦) ∈ 𝑆) | ||
Theorem | cpmatsubgpmat 21328 | The set of all constant polynomial matrices over a ring 𝑅 is an additive subgroup of the ring of all polynomial matrices over the ring 𝑅. (Contributed by AV, 15-Nov-2019.) |
⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubGrp‘𝐶)) | ||
Theorem | cpmatsrgpmat 21329 | The set of all constant polynomial matrices over a ring 𝑅 is a subring of the ring of all polynomial matrices over the ring 𝑅. (Contributed by AV, 18-Nov-2019.) |
⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubRing‘𝐶)) | ||
Theorem | 0elcpmat 21330 | The zero of the ring of all polynomial matrices over the ring 𝑅 is a constant polynomial matrix. (Contributed by AV, 27-Nov-2019.) |
⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g‘𝐶) ∈ 𝑆) | ||
Theorem | mat2pmatfval 21331* | Value of the matrix transformation. (Contributed by AV, 31-Jul-2019.) |
⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑆 = (algSc‘𝑃) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑇 = (𝑚 ∈ 𝐵 ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑆‘(𝑥𝑚𝑦))))) | ||
Theorem | mat2pmatval 21332* | The result of a matrix transformation. (Contributed by AV, 31-Jul-2019.) |
⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑆 = (algSc‘𝑃) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ (𝑆‘(𝑥𝑀𝑦)))) | ||
Theorem | mat2pmatvalel 21333 | A (matrix) element of the result of a matrix transformation. (Contributed by AV, 31-Jul-2019.) |
⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑆 = (algSc‘𝑃) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁)) → (𝑋(𝑇‘𝑀)𝑌) = (𝑆‘(𝑋𝑀𝑌))) | ||
Theorem | mat2pmatbas 21334 | The result of a matrix transformation is a polynomial matrix. (Contributed by AV, 1-Aug-2019.) |
⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ (Base‘𝐶)) | ||
Theorem | mat2pmatbas0 21335 | The result of a matrix transformation is a polynomial matrix. (Contributed by AV, 27-Oct-2019.) |
⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐻 = (Base‘𝐶) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ 𝐻) | ||
Theorem | mat2pmatf 21336 | The matrix transformation is a function from the matrices to the polynomial matrices. (Contributed by AV, 27-Oct-2019.) |
⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐻 = (Base‘𝐶) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵⟶𝐻) | ||
Theorem | mat2pmatf1 21337 | The matrix transformation is a 1-1 function from the matrices to the polynomial matrices. (Contributed by AV, 28-Oct-2019.) (Proof shortened by AV, 27-Nov-2019.) |
⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐻 = (Base‘𝐶) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵–1-1→𝐻) | ||
Theorem | mat2pmatghm 21338 | The transformation of matrices into polynomial matrices is an additive group homomorphism. (Contributed by AV, 28-Oct-2019.) (Proof shortened by AV, 28-Nov-2019.) |
⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐻 = (Base‘𝐶) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐴 GrpHom 𝐶)) | ||
Theorem | mat2pmatmul 21339* | The transformation of matrices into polynomial matrices preserves the multiplication. (Contributed by AV, 29-Oct-2019.) (Proof shortened by AV, 28-Nov-2019.) |
⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐻 = (Base‘𝐶) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑇‘(𝑥(.r‘𝐴)𝑦)) = ((𝑇‘𝑥)(.r‘𝐶)(𝑇‘𝑦))) | ||
Theorem | mat2pmat1 21340 | The transformation of the identity matrix results in the identity polynomial matrix. (Contributed by AV, 29-Oct-2019.) |
⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐻 = (Base‘𝐶) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑇‘(1r‘𝐴)) = (1r‘𝐶)) | ||
Theorem | mat2pmatmhm 21341 | The transformation of matrices into polynomial matrices is a homomorphism of multiplicative monoids. (Contributed by AV, 29-Oct-2019.) |
⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐻 = (Base‘𝐶) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝐶))) | ||
Theorem | mat2pmatrhm 21342 | The transformation of matrices into polynomial matrices is a ring homomorphism. (Contributed by AV, 29-Oct-2019.) |
⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐻 = (Base‘𝐶) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 RingHom 𝐶)) | ||
Theorem | mat2pmatlin 21343 | The transformation of matrices into polynomial matrices is "linear", analogous to lmhmlin 19807. Since 𝐴 and 𝐶 have different scalar rings, 𝑇 cannot be a left module homomorphism as defined in df-lmhm 19794, see lmhmsca 19802. (Contributed by AV, 13-Nov-2019.) (Proof shortened by AV, 28-Nov-2019.) |
⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐻 = (Base‘𝐶) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ 𝑆 = (algSc‘𝑃) & ⊢ · = ( ·𝑠 ‘𝐴) & ⊢ × = ( ·𝑠 ‘𝐶) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵)) → (𝑇‘(𝑋 · 𝑌)) = ((𝑆‘𝑋) × (𝑇‘𝑌))) | ||
Theorem | 0mat2pmat 21344 | The transformed zero matrix is the zero polynomial matrix. (Contributed by AV, 5-Aug-2019.) (Proof shortened by AV, 19-Nov-2019.) |
⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 0 = (0g‘(𝑁 Mat 𝑅)) & ⊢ 𝑍 = (0g‘(𝑁 Mat 𝑃)) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → (𝑇‘ 0 ) = 𝑍) | ||
Theorem | idmatidpmat 21345 | The transformed identity matrix is the identity polynomial matrix. (Contributed by AV, 1-Aug-2019.) (Proof shortened by AV, 19-Nov-2019.) |
⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 1 = (1r‘(𝑁 Mat 𝑅)) & ⊢ 𝐼 = (1r‘(𝑁 Mat 𝑃)) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → (𝑇‘ 1 ) = 𝐼) | ||
Theorem | d0mat2pmat 21346 | The transformed empty set as matrix of dimenson 0 is the empty set (i.e. the polynomial matrix of dimension 0). (Contributed by AV, 4-Aug-2019.) |
⊢ (𝑅 ∈ 𝑉 → ((∅ matToPolyMat 𝑅)‘∅) = ∅) | ||
Theorem | d1mat2pmat 21347 | The transformation of a matrix of dimenson 1. (Contributed by AV, 4-Aug-2019.) |
⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐵 = (Base‘(𝑁 Mat 𝑅)) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑆 = (algSc‘𝑃) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) = {〈〈𝐴, 𝐴〉, (𝑆‘(𝐴𝑀𝐴))〉}) | ||
Theorem | mat2pmatscmxcl 21348 | A transformed matrix multiplied with a power of the variable of a polynomial is a polynomial matrix. (Contributed by AV, 6-Nov-2019.) (Proof shortened by AV, 28-Nov-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐾 = (Base‘𝐴) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ ∗ = ( ·𝑠 ‘𝐶) & ⊢ ↑ = (.g‘(mulGrp‘𝑃)) & ⊢ 𝑋 = (var1‘𝑅) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) → ((𝐿 ↑ 𝑋) ∗ (𝑇‘𝑀)) ∈ 𝐵) | ||
Theorem | m2cpm 21349 | The result of a matrix transformation is a constant polynomial matrix. (Contributed by AV, 18-Nov-2019.) (Proof shortened by AV, 28-Nov-2019.) |
⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) ∈ 𝑆) | ||
Theorem | m2cpmf 21350 | The matrix transformation is a function from the matrices to the constant polynomial matrices. (Contributed by AV, 18-Nov-2019.) |
⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵⟶𝑆) | ||
Theorem | m2cpmf1 21351 | The matrix transformation is a 1-1 function from the matrices to the constant polynomial matrices. (Contributed by AV, 18-Nov-2019.) |
⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵–1-1→𝑆) | ||
Theorem | m2cpmghm 21352 | The transformation of matrices into constant polynomial matrices is an additive group homomorphism. (Contributed by AV, 18-Nov-2019.) |
⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝑈 = (𝐶 ↾s 𝑆) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐴 GrpHom 𝑈)) | ||
Theorem | m2cpmmhm 21353 | The transformation of matrices into constant polynomial matrices is a homomorphism of multiplicative monoids. (Contributed by AV, 18-Nov-2019.) |
⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝑈 = (𝐶 ↾s 𝑆) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝑈))) | ||
Theorem | m2cpmrhm 21354 | The transformation of matrices into constant polynomial matrices is a ring homomorphism. (Contributed by AV, 18-Nov-2019.) |
⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝑈 = (𝐶 ↾s 𝑆) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 RingHom 𝑈)) | ||
Theorem | m2pmfzmap 21355 | The transformed values of a (finite) mapping of integers to matrices. (Contributed by AV, 4-Nov-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑌 = (𝑁 Mat 𝑃) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑆 ∈ ℕ0) ∧ (𝑏 ∈ (𝐵 ↑m (0...𝑆)) ∧ 𝐼 ∈ (0...𝑆))) → (𝑇‘(𝑏‘𝐼)) ∈ (Base‘𝑌)) | ||
Theorem | m2pmfzgsumcl 21356* | Closure of the sum of scaled transformed matrices. (Contributed by AV, 4-Nov-2019.) (Proof shortened by AV, 28-Nov-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝑌 = (𝑁 Mat 𝑃) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝑋 = (var1‘𝑅) & ⊢ ↑ = (.g‘(mulGrp‘𝑃)) & ⊢ · = ( ·𝑠 ‘𝑌) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ (𝑠 ∈ ℕ0 ∧ 𝑏 ∈ (𝐵 ↑m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 ↑ 𝑋) · (𝑇‘(𝑏‘𝑖))))) ∈ (Base‘𝑌)) | ||
Theorem | cpm2mfval 21357* | Value of the inverse matrix transformation. (Contributed by AV, 14-Dec-2019.) |
⊢ 𝐼 = (𝑁 cPolyMatToMat 𝑅) & ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐼 = (𝑚 ∈ 𝑆 ↦ (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0)))) | ||
Theorem | cpm2mval 21358* | The result of an inverse matrix transformation. (Contributed by AV, 12-Nov-2019.) (Revised by AV, 14-Dec-2019.) |
⊢ 𝐼 = (𝑁 cPolyMatToMat 𝑅) & ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝑆) → (𝐼‘𝑀) = (𝑥 ∈ 𝑁, 𝑦 ∈ 𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0))) | ||
Theorem | cpm2mvalel 21359 | A (matrix) element of the result of an inverse matrix transformation. (Contributed by AV, 14-Dec-2019.) |
⊢ 𝐼 = (𝑁 cPolyMatToMat 𝑅) & ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝑆) ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁)) → (𝑋(𝐼‘𝑀)𝑌) = ((coe1‘(𝑋𝑀𝑌))‘0)) | ||
Theorem | cpm2mf 21360 | The inverse matrix transformation is a function from the constant polynomial matrices to the matrices over the base ring of the polynomials. (Contributed by AV, 24-Nov-2019.) (Revised by AV, 15-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐾 = (Base‘𝐴) & ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) & ⊢ 𝐼 = (𝑁 cPolyMatToMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐼:𝑆⟶𝐾) | ||
Theorem | m2cpminvid 21361 | The inverse transformation applied to the transformation of a matrix over a ring R results in the matrix itself. (Contributed by AV, 12-Nov-2019.) (Revised by AV, 13-Dec-2019.) |
⊢ 𝐼 = (𝑁 cPolyMatToMat 𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐾 = (Base‘𝐴) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) → (𝐼‘(𝑇‘𝑀)) = 𝑀) | ||
Theorem | m2cpminvid2lem 21362* | Lemma for m2cpminvid2 21363. (Contributed by AV, 12-Nov-2019.) (Revised by AV, 14-Dec-2019.) |
⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) ⇒ ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → ∀𝑛 ∈ ℕ0 ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛)) | ||
Theorem | m2cpminvid2 21363 | The transformation applied to the inverse transformation of a constant polynomial matrix over the ring 𝑅 results in the matrix itself. (Contributed by AV, 12-Nov-2019.) (Revised by AV, 14-Dec-2019.) |
⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) & ⊢ 𝐼 = (𝑁 cPolyMatToMat 𝑅) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝑆) → (𝑇‘(𝐼‘𝑀)) = 𝑀) | ||
Theorem | m2cpmfo 21364 | The matrix transformation is a function from the matrices onto the constant polynomial matrices. (Contributed by AV, 19-Nov-2019.) (Proof shortened by AV, 28-Nov-2019.) |
⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐾 = (Base‘𝐴) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐾–onto→𝑆) | ||
Theorem | m2cpmf1o 21365 | The matrix transformation is a 1-1 function from the matrices onto the constant polynomial matrices. (Contributed by AV, 19-Nov-2019.) |
⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐾 = (Base‘𝐴) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐾–1-1-onto→𝑆) | ||
Theorem | m2cpmrngiso 21366 | The transformation of matrices into constant polynomial matrices is a ring isomorphism. (Contributed by AV, 19-Nov-2019.) |
⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) & ⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐾 = (Base‘𝐴) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝑈 = (𝐶 ↾s 𝑆) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 RingIso 𝑈)) | ||
Theorem | matcpmric 21367 | The ring of matrices over a commutative ring is isomorphic to the ring of scalar matrices over the same ring. (Contributed by AV, 30-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) & ⊢ 𝑈 = (𝐶 ↾s 𝑆) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ≃𝑟 𝑈) | ||
Theorem | m2cpminv 21368 | The inverse matrix transformation is a 1-1 function from the constant polynomial matrices onto the matrices over the base ring of the polynomials. (Contributed by AV, 27-Nov-2019.) (Revised by AV, 15-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐾 = (Base‘𝐴) & ⊢ 𝑆 = (𝑁 ConstPolyMat 𝑅) & ⊢ 𝐼 = (𝑁 cPolyMatToMat 𝑅) & ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝐼:𝑆–1-1-onto→𝐾 ∧ ◡𝐼 = 𝑇)) | ||
Theorem | m2cpminv0 21369 | The inverse matrix transformation applied to the zero polynomial matrix results in the zero of the matrices over the base ring of the polynomials. (Contributed by AV, 24-Nov-2019.) (Revised by AV, 15-Dec-2019.) |
⊢ 𝐴 = (𝑁 Mat 𝑅) & ⊢ 𝐼 = (𝑁 cPolyMatToMat 𝑅) & ⊢ 𝑃 = (Poly1‘𝑅) & ⊢ 𝐶 = (𝑁 Mat 𝑃) & ⊢ 0 = (0g‘𝐴) & ⊢ 𝑍 = (0g‘𝐶) ⇒ ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝐼‘𝑍) = 0 ) | ||
In this section, the decomposition of polynomial matrices into (polynomial) multiples of constant (polynomial) matrices is prepared by collecting the coefficients of a polynomial matrix which belong to the same power of the polynomial variable. Such a collection is given by the functiondecompPMat (see df-decpmat 21371), which maps a polynomial matrix 𝑀 to a constant matrix consisting of the coefficients of the scaled monomials ((𝑐‘𝑘) ∗ (𝑘 ↑ 𝑋)), i.e. the coefficients belonging to the k-th power of the polynomial variable 𝑋, of each entry in the polynomial matrix 𝑀. The resulting decomposition is provided by theorem pmatcollpw 21389. | ||
Syntax | cdecpmat 21370 | Extend class notation to include the decomposition of polynomial matrices. |
class decompPMat | ||
Definition | df-decpmat 21371* | 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 21372* | 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 21373* | 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 21374 | 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 21375 | 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 21376* | 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 21377* | 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 21378 | 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 21379* | Lemma for decpmatmul 21380. (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 21380* | 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 21381* | Lemma 0 for pm2mpmhm 21428. (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 21382* | Lemma 1 for pmatcollpw1 21384. (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 21383* | Lemma 2 for pmatcollpw1 21384: 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 21384* | 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 21385* | Lemma for pmatcollpw2 21386. (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 21386* | 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 21387 | 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 21378 (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 21388 | Lemma for pmatcollpw 21389. (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 21389* | 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 21390* | 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 21391* | Lemma for pmatcollpw3 21392 and pmatcollpw3fi 21393: 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 21392* | 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 21393* | 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 21394* | Lemma 1 for pmatcollpw3fi1 21396. (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 21395* | Lemma 2 for pmatcollpw3fi1 21396. (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 21396* | 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 21397 | Lemma 1 for pmatcollpwscmat 21399. (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 21398 | Lemma 2 for pmatcollpwscmat 21399. (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 21399* | 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 21431, 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 21400 | Extend class notation with the transformation of a polynomial matrix into a polynomial over matrices. |
class pMatToMatPoly |
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