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Theorem List for Metamath Proof Explorer - 20501-20600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem1pmatscmul 20501 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &   𝐸 = (Base‘𝑃)    &    = ( ·𝑠𝐶)    &    1 = (1r𝐶)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑄𝐸) → (𝑄 1 ) ∈ 𝐵)
 
11.4.2  Constant polynomial matrices

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 19308) to a scalar 𝐴𝑅: ((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 20506), 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 20520) 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 20557). 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 20558. 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 20559.

 
Syntaxccpmat 20502 Extend class notation with the set of all constant polynomial matrices.
class ConstPolyMat
 
Syntaxcmat2pmat 20503 Extend class notation with the transformation of a matrix into a matrix of polynomials.
class matToPolyMat
 
Syntaxccpmat2mat 20504 Extend class notation with the transformation of a constant polynomial matrix into a matrix.
class cPolyMatToMat
 
Definitiondf-cpmat 20505* The set of all constant polynomial matrices, which are all matrices whose entries are constant polynomials (or "scalar polynomials", see ply1sclf 19649). (Contributed by AV, 15-Nov-2019.)
ConstPolyMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ {𝑚 ∈ (Base‘(𝑛 Mat (Poly1𝑟))) ∣ ∀𝑖𝑛𝑗𝑛𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑟)})
 
Definitiondf-mat2pmat 20506* 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)) ↦ (𝑥𝑛, 𝑦𝑛 ↦ ((algSc‘(Poly1𝑟))‘(𝑥𝑚𝑦)))))
 
Definitiondf-cpmat2mat 20507* 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 20559, 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) ↦ (𝑥𝑛, 𝑦𝑛 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
 
Theoremcpmat 20508* 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑆 = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑚𝑗))‘𝑘) = (0g𝑅)})
 
Theoremcpmatpmat 20509 A constant polynomial matrix is a polynomial matrix. (Contributed by AV, 16-Nov-2019.)
𝑆 = (𝑁 ConstPolyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)       ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝑆) → 𝑀𝐵)
 
Theoremcpmatel 20510* Property of a constant polynomial matrix. (Contributed by AV, 15-Nov-2019.)
𝑆 = (𝑁 ConstPolyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)       ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → (𝑀𝑆 ↔ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g𝑅)))
 
Theoremcpmatelimp 20511* Implication of a set being a constant polynomial matrix. (Contributed by AV, 18-Nov-2019.)
𝑆 = (𝑁 ConstPolyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑀𝑆 → (𝑀𝐵 ∧ ∀𝑖𝑁𝑗𝑁𝑘 ∈ ℕ ((coe1‘(𝑖𝑀𝑗))‘𝑘) = (0g𝑅))))
 
Theoremcpmatel2 20512* 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &   𝐾 = (Base‘𝑅)    &   𝐴 = (algSc‘𝑃)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑀𝑆 ↔ ∀𝑖𝑁𝑗𝑁𝑘𝐾 (𝑖𝑀𝑗) = (𝐴𝑘)))
 
Theoremcpmatelimp2 20513* Another implication of a set being a constant polynomial matrix. (Contributed by AV, 17-Nov-2019.)
𝑆 = (𝑁 ConstPolyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &   𝐾 = (Base‘𝑅)    &   𝐴 = (algSc‘𝑃)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑀𝑆 → (𝑀𝐵 ∧ ∀𝑖𝑁𝑗𝑁𝑘𝐾 (𝑖𝑀𝑗) = (𝐴𝑘))))
 
Theorem1elcpmat 20514 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r𝐶) ∈ 𝑆)
 
Theoremcpmatacl 20515* 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝐶)𝑦) ∈ 𝑆)
 
Theoremcpmatinvcl 20516* 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥𝑆 ((invg𝐶)‘𝑥) ∈ 𝑆)
 
Theoremcpmatmcllem 20517* Lemma for cpmatmcl 20518. (Contributed by AV, 18-Nov-2019.)
𝑆 = (𝑁 ConstPolyMat 𝑅)    &   𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝑆𝑦𝑆)) → ∀𝑖𝑁𝑗𝑁𝑐 ∈ ℕ ((coe1‘(𝑃 Σg (𝑘𝑁 ↦ ((𝑖𝑥𝑘)(.r𝑃)(𝑘𝑦𝑗)))))‘𝑐) = (0g𝑅))
 
Theoremcpmatmcl 20518* 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑥𝑆𝑦𝑆 (𝑥(.r𝐶)𝑦) ∈ 𝑆)
 
Theoremcpmatsubgpmat 20519 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubGrp‘𝐶))
 
Theoremcpmatsrgpmat 20520 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubRing‘𝐶))
 
Theorem0elcpmat 20521 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g𝐶) ∈ 𝑆)
 
Theoremmat2pmatfval 20522* Value of the matrix transformation. (Contributed by AV, 31-Jul-2019.)
𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑆 = (algSc‘𝑃)       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑇 = (𝑚𝐵 ↦ (𝑥𝑁, 𝑦𝑁 ↦ (𝑆‘(𝑥𝑚𝑦)))))
 
Theoremmat2pmatval 20523* The result of a matrix transformation. (Contributed by AV, 31-Jul-2019.)
𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Poly1𝑅)    &   𝑆 = (algSc‘𝑃)       ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) → (𝑇𝑀) = (𝑥𝑁, 𝑦𝑁 ↦ (𝑆‘(𝑥𝑀𝑦))))
 
Theoremmat2pmatvalel 20524 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 = (Poly1𝑅)    &   𝑆 = (algSc‘𝑃)       (((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝐵) ∧ (𝑋𝑁𝑌𝑁)) → (𝑋(𝑇𝑀)𝑌) = (𝑆‘(𝑋𝑀𝑌)))
 
Theoremmat2pmatbas 20525 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑇𝑀) ∈ (Base‘𝐶))
 
Theoremmat2pmatbas0 20526 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐻 = (Base‘𝐶)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑇𝑀) ∈ 𝐻)
 
Theoremmat2pmatf 20527 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐻 = (Base‘𝐶)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵𝐻)
 
Theoremmat2pmatf1 20528 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐻 = (Base‘𝐶)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵1-1𝐻)
 
Theoremmat2pmatghm 20529 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐻 = (Base‘𝐶)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐴 GrpHom 𝐶))
 
Theoremmat2pmatmul 20530* 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐻 = (Base‘𝐶)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → ∀𝑥𝐵𝑦𝐵 (𝑇‘(𝑥(.r𝐴)𝑦)) = ((𝑇𝑥)(.r𝐶)(𝑇𝑦)))
 
Theoremmat2pmat1 20531 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐻 = (Base‘𝐶)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑇‘(1r𝐴)) = (1r𝐶))
 
Theoremmat2pmatmhm 20532 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐻 = (Base‘𝐶)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝐶)))
 
Theoremmat2pmatrhm 20533 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐻 = (Base‘𝐶)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 RingHom 𝐶))
 
Theoremmat2pmatlin 20534 The transformation of matrices into polynomial matrices is "linear", analogous to lmhmlin 19029. Since 𝐴 and 𝐶 have different scalar rings, 𝑇 cannot be a left module homomorphism as defined in df-lmhm 19016, see lmhmsca 19024. (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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐻 = (Base‘𝐶)    &   𝐾 = (Base‘𝑅)    &   𝑆 = (algSc‘𝑃)    &    · = ( ·𝑠𝐴)    &    × = ( ·𝑠𝐶)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑋𝐾𝑌𝐵)) → (𝑇‘(𝑋 · 𝑌)) = ((𝑆𝑋) × (𝑇𝑌)))
 
Theorem0mat2pmat 20535 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 = (Poly1𝑅)    &    0 = (0g‘(𝑁 Mat 𝑅))    &   𝑍 = (0g‘(𝑁 Mat 𝑃))       ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → (𝑇0 ) = 𝑍)
 
Theoremidmatidpmat 20536 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 = (Poly1𝑅)    &    1 = (1r‘(𝑁 Mat 𝑅))    &   𝐼 = (1r‘(𝑁 Mat 𝑃))       ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → (𝑇1 ) = 𝐼)
 
Theoremd0mat2pmat 20537 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.)
(𝑅𝑉 → ((∅ matToPolyMat 𝑅)‘∅) = ∅)
 
Theoremd1mat2pmat 20538 The transformation of a matrix of dimenson 1. (Contributed by AV, 4-Aug-2019.)
𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝐵 = (Base‘(𝑁 Mat 𝑅))    &   𝑃 = (Poly1𝑅)    &   𝑆 = (algSc‘𝑃)       ((𝑅𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴𝑉) ∧ 𝑀𝐵) → (𝑇𝑀) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩})
 
Theoremmat2pmatscmxcl 20539 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝐶)    &    = (.g‘(mulGrp‘𝑃))    &   𝑋 = (var1𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀𝐾𝐿 ∈ ℕ0)) → ((𝐿 𝑋) (𝑇𝑀)) ∈ 𝐵)
 
Theoremm2cpm 20540 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 ∧ 𝑀𝐵) → (𝑇𝑀) ∈ 𝑆)
 
Theoremm2cpmf 20541 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:𝐵𝑆)
 
Theoremm2cpmf1 20542 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:𝐵1-1𝑆)
 
Theoremm2cpmghm 20543 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝑈 = (𝐶s 𝑆)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐴 GrpHom 𝑈))
 
Theoremm2cpmmhm 20544 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝑈 = (𝐶s 𝑆)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ ((mulGrp‘𝐴) MndHom (mulGrp‘𝑈)))
 
Theoremm2cpmrhm 20545 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝑈 = (𝐶s 𝑆)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 RingHom 𝑈))
 
Theoremm2pmfzmap 20546 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 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝑇 = (𝑁 matToPolyMat 𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑆 ∈ ℕ0) ∧ (𝑏 ∈ (𝐵𝑚 (0...𝑆)) ∧ 𝐼 ∈ (0...𝑆))) → (𝑇‘(𝑏𝐼)) ∈ (Base‘𝑌))
 
Theoremm2pmfzgsumcl 20547* 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 = (Poly1𝑅)    &   𝑌 = (𝑁 Mat 𝑃)    &   𝑇 = (𝑁 matToPolyMat 𝑅)    &   𝑋 = (var1𝑅)    &    = (.g‘(mulGrp‘𝑃))    &    · = ( ·𝑠𝑌)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵𝑚 (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 𝑋) · (𝑇‘(𝑏𝑖))))) ∈ (Base‘𝑌))
 
Theoremcpm2mfval 20548* Value of the inverse matrix transformation. (Contributed by AV, 14-Dec-2019.)
𝐼 = (𝑁 cPolyMatToMat 𝑅)    &   𝑆 = (𝑁 ConstPolyMat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝐼 = (𝑚𝑆 ↦ (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑚𝑦))‘0))))
 
Theoremcpm2mval 20549* 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 ∧ 𝑅𝑉𝑀𝑆) → (𝐼𝑀) = (𝑥𝑁, 𝑦𝑁 ↦ ((coe1‘(𝑥𝑀𝑦))‘0)))
 
Theoremcpm2mvalel 20550 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 ∧ 𝑅𝑉𝑀𝑆) ∧ (𝑋𝑁𝑌𝑁)) → (𝑋(𝐼𝑀)𝑌) = ((coe1‘(𝑋𝑀𝑌))‘0))
 
Theoremcpm2mf 20551 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:𝑆𝐾)
 
Theoremm2cpminvid 20552 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 ∧ 𝑀𝐾) → (𝐼‘(𝑇𝑀)) = 𝑀)
 
Theoremm2cpminvid2lem 20553* Lemma for m2cpminvid2 20554. (Contributed by AV, 12-Nov-2019.) (Revised by AV, 14-Dec-2019.)
𝑆 = (𝑁 ConstPolyMat 𝑅)    &   𝑃 = (Poly1𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝑥𝑁𝑦𝑁)) → ∀𝑛 ∈ ℕ0 ((coe1‘((algSc‘𝑃)‘((coe1‘(𝑥𝑀𝑦))‘0)))‘𝑛) = ((coe1‘(𝑥𝑀𝑦))‘𝑛))
 
Theoremm2cpminvid2 20554 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 ∧ 𝑀𝑆) → (𝑇‘(𝐼𝑀)) = 𝑀)
 
Theoremm2cpmfo 20555 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:𝐾onto𝑆)
 
Theoremm2cpmf1o 20556 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:𝐾1-1-onto𝑆)
 
Theoremm2cpmrngiso 20557 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝑈 = (𝐶s 𝑆)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 RingIso 𝑈))
 
Theoremmatcpmric 20558 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝑆 = (𝑁 ConstPolyMat 𝑅)    &   𝑈 = (𝐶s 𝑆)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴𝑟 𝑈)
 
Theoremm2cpminv 20559 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:𝑆1-1-onto𝐾𝐼 = 𝑇))
 
Theoremm2cpminv0 20560 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 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &    0 = (0g𝐴)    &   𝑍 = (0g𝐶)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝐼𝑍) = 0 )
 
11.4.3  Collecting coefficients of polynomial matrices

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 20562), which maps a polynomial matrix 𝑀 to a constant matrix consisting of the coefficients of the scaled monomials ((𝑐𝑘) (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 20580.

 
Syntaxcdecpmat 20561 Extend class notation to include the decomposition of polynomial matrices.
class decompPMat
 
Definitiondf-decpmat 20562* 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 20563* 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 20564* 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 20565 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 20566 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 20567* 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 20568* 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 20569 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 20570* Lemma for decpmatmul 20571. (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 20571* 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 20572* Lemma 0 for pm2mpmhm 20619. (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 20573* Lemma 1 for pmatcollpw1 20575. (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 20574* Lemma 2 for pmatcollpw1 20575: 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 20575* 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 20576* Lemma for pmatcollpw2 20577. (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 20577* 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 20578 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 20569 (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 20579 Lemma for pmatcollpw 20580. (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 20580* 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 20581* Write a polynomial matrix (over a commutative ring) as a finite sum of products of variable powers and constant matrices with scalar entries. (Contributed by AV, 4-Nov-2019.) (Revised by AV, 4-Dec-2019.)
𝑃 = (Poly1𝑅)    &   𝐶 = (𝑁 Mat 𝑃)    &   𝐵 = (Base‘𝐶)    &    = ( ·𝑠𝐶)    &    = (.g‘(mulGrp‘𝑃))    &   𝑋 = (var1𝑅)    &   𝑇 = (𝑁 matToPolyMat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ∃𝑠 ∈ ℕ0 𝑀 = (𝐶 Σg (𝑛 ∈ (0...𝑠) ↦ ((𝑛 𝑋) (𝑇‘(𝑀 decompPMat 𝑛))))))
 
Theorempmatcollpw3lem 20582* Lemma for pmatcollpw3 20583 and pmatcollpw3fi 20584: 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 20583* 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 20584* 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 20585* Lemma 1 for pmatcollpw3fi1 20587. (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 20586* Lemma 2 for pmatcollpw3fi1 20587. (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 20587* 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 20588 Lemma 1 for pmatcollpwscmat 20590. (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 20589 Lemma 2 for pmatcollpwscmat 20590. (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 20590* 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 20622, 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 20614 and pm2mprngiso 20621), 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 20610).

In this section, the following conventions are mostly used:

  • 𝑅 is a (unital) ring (see df-ring 18543)
  • 𝑃 = (Poly1𝑅) is the polynomial algebra over (the ring) 𝑅 (see df-ply1 19546)
    • 𝐾 = (Base‘𝑃) is its base set (see df-base 15857)
    • 𝑌 = (var1𝑅) is its variable (see df-vr1 19545)
    • · = ( ·𝑠𝑃) is its scalar multiplication (see df-vsca 15952 or lmodvscl 18874)
    • 𝐸 = (.g‘(mulGrp‘𝑃)) is its exponentiation (see df-mulg 17535)
  • 𝐴 = (𝑁 Mat 𝑅) is the algebra of N x N matrices over (the ring) 𝑅 (see df-mat 20208)
  • 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 20591 Extend class notation with the transformation of a polynomial matrix into a polynomial over matrices.
class pMatToMatPoly
 
Definitiondf-pm2mp 20592* 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 20593* Lemma for pm2mpf1 20598. (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 20594* 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 20595* 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 20596 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 20597 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 20598 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 20599 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 20600 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))
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