| import Mathlib |
| import OAI.Probability.SKGap.Localization.RealPart |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace SKGap.ComplexMatrix |
| open Matrix |
| open scoped Matrix.Norms.Frobenius SchwartzMap |
| variable {ι : Type*} [Fintype ι] [DecidableEq ι] |
|
|
| lemma conjugate_frobenius (U : unitary (Matrix ι ι ℂ)) (M : Matrix ι ι ℂ) : |
| ‖conjugate U M‖ = ‖M‖ := by |
| change ‖(U : Matrix ι ι ℂ)*M*(U : Matrix ι ι ℂ)ᴴ‖ = _ |
| rw [frobenius_mul_unitary ((U : Matrix ι ι ℂ)*M) ((U : Matrix ι ι ℂ)ᴴ) (Unitary.star_mem U.property), |
| frobenius_unitary_mul _ _ U.property] |
|
|
| noncomputable def conjugateIsometry (U : unitary (Matrix ι ι ℂ)) : |
| Matrix ι ι ℂ ≃ₗᵢ[ℝ] Matrix ι ι ℂ where |
| toLinearEquiv := (conjugate U).toAlgEquiv.toLinearEquiv.restrictScalars ℝ |
| norm_map' := conjugate_frobenius U |
|
|
| lemma conjugate_opNorm (U : unitary (Matrix ι ι ℂ)) (M : Matrix ι ι ℂ) : |
| opNorm (conjugate U M) = opNorm M := by |
| change ‖lin ((U : Matrix ι ι ℂ)*M*(U : Matrix ι ι ℂ)ᴴ)‖ = _ |
| rw [lin_mul,lin_mul] |
| have hstar : lin ((U : Matrix ι ι ℂ)ᴴ) ∈ unitary _ := lin_unitary _ (Unitary.star_mem U.property) |
| rw [CStarRing.norm_mul_mem_unitary _ hstar, |
| CStarRing.norm_mem_unitary_mul _ (lin_unitary _ U.property)] |
| rfl |
|
|
| lemma matrixProject_conjugate (U : unitary (Matrix ι ι ℂ)) {R : ℝ} (hR : 0 ≤ R) |
| (M : Matrix ι ι ℂ) : |
| matrixProject R hR (conjugate U M) = conjugate U (matrixProject R hR M) := by |
| apply convexProject_equivariant (hermitianBall_good hR) (conjugateIsometry U) |
| intro N |
| change ((conjugate U N)ᴴ = conjugate U N ∧ opNorm (conjugate U N) ≤ R) ↔ |
| (Nᴴ = N ∧ opNorm N ≤ R) |
| have hh : (conjugate U N)ᴴ = conjugate U Nᴴ := (map_star (conjugate U) N).symm |
| have he : conjugate U Nᴴ = conjugate U N ↔ Nᴴ=N := |
| ⟨fun h => (conjugate U).injective h,fun h => congrArg (conjugate U) h⟩ |
| rw [hh,he,conjugate_opNorm] |
|
|
| lemma truncatedG_conjugate (f : 𝓢(ℝ,ℂ)) (U : unitary (Matrix ι ι ℂ)) {R : ℝ} (hR : 0 ≤ R) |
| (B D M : Matrix ι ι ℂ) (hB : Bᴴ=B) (hD : Dᴴ=D) : |
| truncatedG f R hR (conjugate U B) (conjugate U D) (conjugate U M) = |
| conjugate U (truncatedG f R hR B D M) := by |
| have hs : sandShift (conjugate U B) (conjugate U D) (matrixProject R hR (conjugate U M)) = |
| conjugate U (sandShift B D (matrixProject R hR M)) := by |
| simp only [sandShift,matrixProject_conjugate,map_sub,map_mul] |
| unfold truncatedG |
| rw [hs,schwartzMatrix_conjugate f U _ |
| (sandShift_hermitian _ _ _ hB hD (matrixProject_bound R hR M).1)] |
| simp only [map_mul] |
|
|
| lemma truncatedK_conjugate (f : 𝓢(ℝ,ℂ)) (U : unitary (Matrix ι ι ℂ)) {R : ℝ} (hR : 0 ≤ R) |
| (B D C M : Matrix ι ι ℂ) (hB : Bᴴ=B) (hD : Dᴴ=D) : |
| truncatedK f R hR (conjugate U B) (conjugate U D) (conjugate U C) (conjugate U M) = |
| conjugate U (truncatedK f R hR B D C M) := by |
| simp only [truncatedK,truncatedG_conjugate f U hR B D M hB hD,matrixProject_conjugate, |
| map_add,map_one,map_mul,map_sub] |
| end SKGap.ComplexMatrix |
| end |
| end |
|
|
| section |
| noncomputable section |
| namespace SKGap |
| open Matrix MeasureTheory ProbabilityTheory |
| open RealComplex ComplexMatrix |
| open scoped Matrix.Norms.Frobenius SchwartzMap |
| variable {ι : Type*} [Fintype ι] [DecidableEq ι] |
|
|
| def signConjugate (s : ι → ℝ) (M : Matrix ι ι ℝ) : Matrix ι ι ℝ := |
| fun i k => s i*s k*M i k |
|
|
| noncomputable def signUnitary (s : ι → ℝ) (hs : ∀ i, s i^2=1) : |
| unitary (Matrix ι ι ℂ) := |
| ⟨diagonal (fun i => (s i : ℂ)), by |
| have hh : ∀ i, (s i : ℂ)*(s i : ℂ)=1 := by |
| intro i |
| exact_mod_cast (show s i*s i=1 by nlinarith [hs i]) |
| change (diagonal (fun i => (s i : ℂ)))ᴴ*diagonal (fun i => (s i : ℂ))=1 ∧ |
| diagonal (fun i => (s i : ℂ))*(diagonal (fun i => (s i : ℂ)))ᴴ=1 |
| simp only [diagonal_conjTranspose,Pi.star_apply,Complex.star_def,Complex.conj_ofReal, |
| diagonal_mul_diagonal,hh,diagonal_one,and_self]⟩ |
|
|
| lemma sign_conjugate_entry (s : ι → ℝ) (hs : ∀ i, s i^2=1) (M : Matrix ι ι ℂ) (i k : ι) : |
| conjugate (signUnitary s hs) M i k = (s i:ℂ)*(s k:ℂ)*M i k := by |
| change (diagonal (fun b => (s b:ℂ))*M*(diagonal (fun b => (s b:ℂ)))ᴴ) i k = _ |
| simp only [diagonal_conjTranspose,Pi.star_apply,Complex.star_def,Complex.conj_ofReal,mul_diagonal,diagonal_mul] |
| ring |
|
|
| lemma sign_conjugate_diagonal (s : ι → ℝ) (hs : ∀ i, s i^2=1) (d : ι → ℝ) : |
| conjugate (signUnitary s hs) (liftMatrix (diagonal d)) = liftMatrix (diagonal d) := by |
| ext i k |
| rw [sign_conjugate_entry] |
| by_cases hik : i=k |
| · subst k |
| simp only [liftMatrix,Matrix.map_apply,diagonal_apply_eq] |
| have h : (s i:ℂ)*(s i:ℂ)=1 := by exact_mod_cast (show s i*s i=1 by nlinarith [hs i]) |
| rw [h,one_mul] |
| · simp [liftMatrix,Matrix.diagonal,hik] |
|
|
| lemma sign_conjugate_lift (s : ι → ℝ) (hs : ∀ i, s i^2=1) (M : Matrix ι ι ℝ) : |
| conjugate (signUnitary s hs) (liftMatrix M) = liftMatrix (signConjugate s M) := by |
| ext i k |
| simp [sign_conjugate_entry,liftMatrix,signConjugate] |
|
|
| lemma realPart_sign_conjugate (s : ι → ℝ) (hs : ∀ i, s i^2=1) (M : Matrix ι ι ℂ) : |
| RealComplex.realPart (conjugate (signUnitary s hs) M) = signConjugate s (RealComplex.realPart M) := by |
| ext i k |
| simp [RealComplex.realPart,sign_conjugate_entry,signConjugate,Complex.mul_re] |
|
|
| lemma realTruncatedK_sign (f : 𝓢(ℝ,ℂ)) {R : ℝ} (hR : 0 ≤ R) |
| (s : ι → ℝ) (hs : ∀ i, s i^2=1) (d c : ι → ℝ) (M : Matrix ι ι ℝ) : |
| realTruncatedK f R hR (diagonal d) (diagonal c) (signConjugate s M) = |
| signConjugate s (realTruncatedK f R hR (diagonal d) (diagonal c) M) := by |
| unfold realTruncatedK |
| rw [← sign_conjugate_lift s hs M] |
| have hd := liftMatrix_hermitian (diagonal d) (diagonal_transpose _) |
| have hc := liftMatrix_hermitian (diagonal c) (diagonal_transpose _) |
| have hB : (1+liftMatrix (diagonal d)*liftMatrix (diagonal c)*liftMatrix (diagonal d))ᴴ = |
| 1+liftMatrix (diagonal d)*liftMatrix (diagonal c)*liftMatrix (diagonal d) := by |
| simp only [conjTranspose_add,conjTranspose_one,conjTranspose_mul,hd,hc,mul_assoc] |
| have he := truncatedK_conjugate f (signUnitary s hs) hR |
| (1+liftMatrix (diagonal d)*liftMatrix (diagonal c)*liftMatrix (diagonal d)) |
| (liftMatrix (diagonal d)) (liftMatrix (diagonal c)) (liftMatrix M) hB hd |
| simp only [map_add,map_one,map_mul,sign_conjugate_diagonal] at he |
| rw [he,realPart_sign_conjugate] |
| end SKGap |
| end |
| end |
|
|
| end OAI |
|
|
