File size: 2,847 Bytes
67aee1d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 | import Mathlib
import OAI.Analysis.FreeWords.MatrixConditionalExpectation
import OAI.Analysis.FreeWords.MatrixTraceUnique
namespace OAI
noncomputable section
namespace KKBlockMatrix
open KKMatrixExpectation
noncomputable section
variable {A : Type*} [Ring A] [StarRing A] [Algebra ℂ A] [StarModule ℂ A]
variable {ι κ : Type*} [Fintype ι] [DecidableEq ι] [Fintype κ] [DecidableEq κ]
def blockHom (ρ : Matrix κ κ ℂ →⋆ₐ[ℂ] A) :
Matrix (ι×κ) (ι×κ) ℂ →⋆ₐ[ℂ] Matrix ι ι A :=
{ (ρ.toAlgHom.mapMatrix).comp (Matrix.compAlgEquiv ι κ ℂ ℂ).symm.toAlgHom with
map_star' := by
intro X
ext i j
exact map_star ρ (fun a b => X (j,a) (i,b)) }
omit [StarModule ℂ A] in
@[simp] theorem blockHom_apply (ρ : Matrix κ κ ℂ →⋆ₐ[ℂ] A)
(X : Matrix (ι×κ) (ι×κ) ℂ) (i j : ι) :
blockHom ρ X i j=ρ (fun a b => X (i,a) (j,b)) := rfl
omit [StarModule ℂ A] in
theorem blockHom_injective (ρ : Matrix κ κ ℂ →⋆ₐ[ℂ] A) (hρ : Function.Injective ρ) :
Function.Injective (blockHom (ι := ι) ρ) := by
intro X Y he
ext ⟨i,a⟩ ⟨j,b⟩
exact congrFun (congrFun (hρ (congrFun (congrFun he i) j)) a) b
theorem blockHom_entry_mem (ρ : Matrix κ κ ℂ →⋆ₐ[ℂ] A)
(X : Matrix (ι×κ) (ι×κ) ℂ) (i j : ι) : blockHom ρ X i j∈ρ.range :=
⟨fun a b => X (i,a) (j,b),rfl⟩
theorem blockHom_kronecker {n d : ℕ} (ρ : Matrix (Fin d) (Fin d) ℂ →⋆ₐ[ℂ] A)
(b : Matrix (Fin n) (Fin n) ℂ) :
blockHom ρ (Matrix.kronecker b 1)=numericalHom (A := A) n b := by
ext i j
change ρ (fun a c => b i j*(1 : Matrix (Fin d) (Fin d) ℂ) a c)=algebraMap ℂ A (b i j)
have he : (fun a c => b i j*(1 : Matrix (Fin d) (Fin d) ℂ) a c)=
algebraMap ℂ (Matrix (Fin d) (Fin d) ℂ) (b i j) := by
ext a c
simp [Matrix.algebraMap_eq_diagonal,Matrix.diagonal_apply,Matrix.one_apply]
rw [he]
exact ρ.commutes (b i j)
omit [StarModule ℂ A] in
theorem blockHom_trace {n d : ℕ} (ρ : Matrix (Fin d) (Fin d) ℂ →⋆ₐ[ℂ] A)
[NeZero d] (τ : A →ₗ[ℂ] ℂ) (hτ1 : τ 1=1) (htr : ∀x y,τ (x*y)=τ (y*x))
(X : Matrix (Fin n×Fin d) (Fin n×Fin d) ℂ) :
matrixTraceMap τ n (blockHom ρ X)=((n*d:ℕ):ℂ)⁻¹*Matrix.trace X := by
change (n:ℂ)⁻¹*(∑i:Fin n,τ (ρ (fun a b => X (i,a) (i,b))))=_
have he : (∑i:Fin n,τ (ρ (fun a b => X (i,a) (i,b))))=
∑i:Fin n,(d:ℂ)⁻¹*(∑a:Fin d,X (i,a) (i,a)) := by
apply Finset.sum_congr rfl
intro i hi
simpa only [Fintype.card_fin,Matrix.trace,Matrix.diag] using KKVariableUltra.matrixHom_trace ρ τ hτ1 htr (fun a b => X (i,a) (i,b))
rw [he,← Finset.mul_sum]
simp only [Matrix.trace,Matrix.diag,Fintype.sum_prod_type,Nat.cast_mul,mul_inv_rev]
ring
end
end KKBlockMatrix
end
end OAI
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