refactor: Lint
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jstoobysmith
parent
6a3bb431bf
commit
4fa4a28d5d
3 changed files with 43 additions and 38 deletions
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@ -105,8 +105,8 @@ import HepLean.Mathematics.Fin.Involutions
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import HepLean.Mathematics.LinearMaps
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import HepLean.Mathematics.List
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import HepLean.Mathematics.PiTensorProduct
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import HepLean.Mathematics.SchurTriangulation
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import HepLean.Mathematics.SO3.Basic
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import HepLean.Mathematics.SchurTriangulation
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import HepLean.Meta.AllFilePaths
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import HepLean.Meta.Basic
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import HepLean.Meta.Informal.Basic
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@ -24,14 +24,16 @@ noncomputable def toSelfAdjointMap' (M : ℂ²ˣ²) : ℍ₂ →ₗ[ℝ] ℍ₂
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map_smul' | r, ⟨A, _⟩ => Subtype.ext <| by simp
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open Complex (I normSq) in
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theorem toSelfAdjointMap_det_one' {M : ℂ²ˣ²} (hM : M.IsUpperTriangular) (detM : M.det = 1)
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: LinearMap.det (toSelfAdjointMap' M) = 1 :=
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lemma toSelfAdjointMap_det_one' {M : ℂ²ˣ²} (hM : M.IsUpperTriangular) (detM : M.det = 1) :
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LinearMap.det (toSelfAdjointMap' M) = 1 :=
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let b : Basis (Fin 2 ⊕ Fin 2) ℝ ℍ₂ := Basis.ofEquivFun {
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toFun := fun ⟨A, _⟩ => ![(A 0 0).re, (A 1 1).re] ⊕ᵥ ![(A 0 1).re, (A 0 1).im]
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map_add' := fun _ _ => funext fun | .inl 0 | .inl 1 | .inr 0 | .inr 1 => rfl
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map_smul' := fun _ _ => funext fun | .inl 0 | .inl 1 | .inr 0 | .inr 1 => by simp
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invFun := fun p => {
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val := let z : ℂ := ⟨p (.inr 0), p (.inr 1)⟩ ; !![p (.inl 0), z; conj z, p (.inl 1)]
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val :=
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let z : ℂ := ⟨p (.inr 0), p (.inr 1)⟩
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!![p (.inl 0), z; conj z, p (.inl 1)]
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property := Matrix.ext fun | 0, 0 | 0, 1 | 1, 0 | 1, 1 => by simp
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}
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left_inv := fun ⟨A, hA⟩ => Subtype.ext <| Matrix.ext fun
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@ -46,12 +48,11 @@ theorem toSelfAdjointMap_det_one' {M : ℂ²ˣ²} (hM : M.IsUpperTriangular) (de
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let E₂ : ℂ²ˣ² := !![0, 1; conj 1, 0] -- b (.inr 0)
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let E₃ : ℂ²ˣ² := !![0, I; conj I, 0] -- b (.inr 1)
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let F : Matrix (Fin 2 ⊕ Fin 2) (Fin 2 ⊕ Fin 2) ℝ := LinearMap.toMatrix b b (toSelfAdjointMap' M)
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let A := F.toBlocks₁₁ ; let B := F.toBlocks₁₂ ; let C := F.toBlocks₂₁ ; let D := F.toBlocks₂₂
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let x := M 0 0 ; let y := M 1 1 ; have hM10 : M 1 0 = 0 := hM <| show 0 < 1 by decide
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let A := F.toBlocks₁₁; let B := F.toBlocks₁₂; let C := F.toBlocks₂₁; let D := F.toBlocks₂₂
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let x := M 0 0; let y := M 1 1; have hM10 : M 1 0 = 0 := hM <| show 0 < 1 by decide
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have he : M = !![x, _; 0, y] := Matrix.ext fun | 0, 0 | 0, 1 | 1, 1 => rfl | 1, 0 => hM10
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have he' : Mᴴ = !![conj x, 0; _, conj y] :=
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Matrix.ext fun | 0, 0 | 1, 0 | 1, 1 => rfl | 0, 1 => by simp [hM10]
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have detA_one : normSq x * normSq y = 1 := congrArg Complex.re <|
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calc ↑(normSq x * normSq y)
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_ = x * conj x * (y * conj y) := by simp [Complex.mul_conj]
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@ -64,10 +65,10 @@ theorem toSelfAdjointMap_det_one' {M : ℂ²ˣ²} (hM : M.IsUpperTriangular) (de
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_ = 1 := detM
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have detD_one : D.det = 1 :=
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let z := x * conj y
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have k₀ : (M * E₂ * Mᴴ) 0 1 = z := by rw [he', he] ; simp [E₂]
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have k₀ : (M * E₂ * Mᴴ) 0 1 = z := by rw [he', he]; simp [E₂]
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have k₁ : (M * E₃ * Mᴴ) 0 1 = ⟨-z.im, z.re⟩ :=
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calc
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_ = x * I * conj y := by rw [he', he] ; simp [E₃]
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_ = x * I * conj y := by rw [he', he]; simp [E₃]
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_ = Complex.I * z := by ring
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_ = ⟨-z.im, z.re⟩ := z.I_mul
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have hD : D = !![z.re, -z.im; z.im, z.re] := Matrix.ext fun
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@ -81,33 +82,33 @@ theorem toSelfAdjointMap_det_one' {M : ℂ²ˣ²} (hM : M.IsUpperTriangular) (de
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letI : Invertible D.det := detD_one ▸ invertibleOne
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letI : Invertible D := D.invertibleOfDetInvertible
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have hE : A - B * ⅟D * C = !![normSq x, _; 0, normSq y] :=
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have k : (M * E₀ * Mᴴ) 0 1 = 0 := by rw [he', he] ; simp [E₀]
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have k : (M * E₀ * Mᴴ) 0 1 = 0 := by rw [he', he]; simp [E₀]
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have hC00 : C 0 0 = 0 := congrArg Complex.re k
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have hC10 : C 1 0 = 0 := congrArg Complex.im k
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Matrix.ext fun
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| 0, 1 => rfl
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| 1, 0 =>
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have hA10 : A 1 0 = 0 := congrArg Complex.re <|
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show (M * E₀ * Mᴴ) 1 1 = 0 by rw [he', he] ; simp [E₀]
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show (M * E₀ * Mᴴ) 1 1 = 0 by rw [he', he]; simp [E₀]
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show A 1 0 - (B * ⅟D) 1 ⬝ᵥ (C · 0) = 0 by simp [hC00, hC10, hA10]
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| 0, 0 =>
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have hA00 : A 0 0 = normSq x := congrArg Complex.re <|
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show (M * E₀ * Mᴴ) 0 0 = normSq x by rw [he', he] ; simp [E₀, x.mul_conj]
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show (M * E₀ * Mᴴ) 0 0 = normSq x by rw [he', he]; simp [E₀, x.mul_conj]
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show A 0 0 - (B * ⅟D) 0 ⬝ᵥ (C · 0) = normSq x by simp [hC00, hC10, hA00]
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| 1, 1 =>
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have hA11 : A 1 1 = normSq y := congrArg Complex.re <|
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show (M * E₁ * Mᴴ) 1 1 = normSq y by rw [he', he] ; simp [E₁, y.mul_conj]
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show (M * E₁ * Mᴴ) 1 1 = normSq y by rw [he', he]; simp [E₁, y.mul_conj]
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have hB10 : B 1 0 = 0 := congrArg Complex.re <|
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show (M * E₂ * Mᴴ) 1 1 = 0 by rw [he', he] ; simp [E₂]
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show (M * E₂ * Mᴴ) 1 1 = 0 by rw [he', he]; simp [E₂]
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have hB11 : B 1 1 = 0 := congrArg Complex.re <|
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show (M * E₃ * Mᴴ) 1 1 = 0 by rw [he', he] ; simp [E₃]
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show (M * E₃ * Mᴴ) 1 1 = 0 by rw [he', he]; simp [E₃]
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calc A 1 1 - (B * ⅟D * C) 1 1
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_ = A 1 1 - B 1 ⬝ᵥ ((⅟D * C) · 1) := by noncomm_ring
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_ = normSq y := by simp [hB10, hB11, hA11]
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calc LinearMap.det (toSelfAdjointMap' M)
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_ = F.det := (LinearMap.det_toMatrix ..).symm
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_ = D.det * (A - B * ⅟D * C).det := F.fromBlocks_toBlocks ▸ Matrix.det_fromBlocks₂₂ ..
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_ = 1 := by rw [hE] ; simp [detD_one, detA_one]
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_ = 1 := by rw [hE]; simp [detD_one, detA_one]
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/-- This promotes `Lorentz.SL2C.toSelfAdjointMap M` and its definitional equivalence,
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`Lorentz.SL2C.toSelfAdjointMap' M`, to a linear equivalence by recognising the linear inverse to be
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@ -124,17 +125,17 @@ noncomputable def toSelfAdjointEquiv (M : ℂ²ˣ²) [Invertible M] : ℍ₂ ≃
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_ = M * M⁻¹ * A * (M * M⁻¹)ᴴ := by noncomm_ring [Matrix.conjTranspose_mul]
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_ = A := by simp
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theorem toSelfAdjointMap_mul (M N : ℂ²ˣ²)
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: toSelfAdjointMap' (M * N) = toSelfAdjointMap' M ∘ₗ toSelfAdjointMap' N :=
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lemma toSelfAdjointMap_mul (M N : ℂ²ˣ²) :
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toSelfAdjointMap' (M * N) = toSelfAdjointMap' M ∘ₗ toSelfAdjointMap' N :=
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LinearMap.ext fun A => Subtype.ext <|
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show M * N * A * (M * N)ᴴ = M * (N * A * Nᴴ) * Mᴴ by noncomm_ring [Matrix.conjTranspose_mul]
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theorem toSelfAdjointMap_similar_det (M N : ℂ²ˣ²) [Invertible M]
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: LinearMap.det (toSelfAdjointMap' (M * N * M⁻¹)) = LinearMap.det (toSelfAdjointMap' N) :=
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lemma toSelfAdjointMap_similar_det (M N : ℂ²ˣ²) [Invertible M] :
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LinearMap.det (toSelfAdjointMap' (M * N * M⁻¹)) = LinearMap.det (toSelfAdjointMap' N) :=
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let e := toSelfAdjointEquiv M
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let f := toSelfAdjointMap' N
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suffices toSelfAdjointMap' (M * N * M⁻¹) = e ∘ₗ f ∘ₗ e.symm from this ▸ f.det_conj e
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by rw [toSelfAdjointMap_mul, toSelfAdjointMap_mul] ; rfl
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by rw [toSelfAdjointMap_mul, toSelfAdjointMap_mul]; rfl
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end SL2C
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end Lorentz
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@ -22,10 +22,10 @@ def finAddEquivSigmaCond : Fin (m + n) ≃ Σ b, cond b (Fin m) (Fin n) :=
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variable {i : Fin (m + n)}
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theorem finAddEquivSigmaCond_true (h : i < m) : finAddEquivSigmaCond i = ⟨true, i, h⟩ :=
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lemma finAddEquivSigmaCond_true (h : i < m) : finAddEquivSigmaCond i = ⟨true, i, h⟩ :=
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congrArg sumEquivSigmalCond <| finSumFinEquiv_symm_apply_castAdd ⟨i, h⟩
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theorem finAddEquivSigmaCond_false (h : ¬ i < m) : finAddEquivSigmaCond i = ⟨false, i.subNat' h⟩ :=
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lemma finAddEquivSigmaCond_false (h : ¬ i < m) : finAddEquivSigmaCond i = ⟨false, i.subNat' h⟩ :=
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let j : Fin n := i.subNat' h
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calc finAddEquivSigmaCond i
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_ = finAddEquivSigmaCond (Fin.natAdd m j) :=
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@ -66,15 +66,16 @@ variable [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
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section
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variable [FiniteDimensional 𝕜 E] [Fintype n] [DecidableEq n]
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theorem toMatrixOrthonormal_apply_apply (b : OrthonormalBasis n 𝕜 E) (f : Module.End 𝕜 E) (i j : n)
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: toMatrixOrthonormal b f i j = ⟪b i, f (b j)⟫_𝕜 :=
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lemma toMatrixOrthonormal_apply_apply (b : OrthonormalBasis n 𝕜 E) (f : Module.End 𝕜 E)
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(i j : n) :
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toMatrixOrthonormal b f i j = ⟪b i, f (b j)⟫_𝕜 :=
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calc
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_ = b.repr (f (b j)) i := f.toMatrix_apply ..
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_ = ⟪b i, f (b j)⟫_𝕜 := b.repr_apply_apply ..
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theorem toMatrixOrthonormal_reindex [Fintype m] [DecidableEq m]
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(b : OrthonormalBasis m 𝕜 E) (e : m ≃ n) (f : Module.End 𝕜 E)
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: toMatrixOrthonormal (b.reindex e) f = Matrix.reindex e e (toMatrixOrthonormal b f) :=
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lemma toMatrixOrthonormal_reindex [Fintype m] [DecidableEq m]
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(b : OrthonormalBasis m 𝕜 E) (e : m ≃ n) (f : Module.End 𝕜 E) :
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toMatrixOrthonormal (b.reindex e) f = Matrix.reindex e e (toMatrixOrthonormal b f) :=
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Matrix.ext fun i j =>
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calc toMatrixOrthonormal (b.reindex e) f i j
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_ = (b.reindex e).repr (f (b.reindex e j)) i := f.toMatrix_apply ..
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@ -101,8 +102,8 @@ variable [IsAlgClosed 𝕜]
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/-- **Don't use this definition directly.** This is the key algorithm behind
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`Matrix.schur_triangulation`. -/
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protected noncomputable def SchurTriangulationAux.of
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[NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [FiniteDimensional 𝕜 E] (f : Module.End 𝕜 E)
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: SchurTriangulationAux f :=
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[NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [FiniteDimensional 𝕜 E] (f : Module.End 𝕜 E) :
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SchurTriangulationAux f :=
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haveI : Decidable (Nontrivial E) := Classical.propDecidable _
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if hE : Nontrivial E then
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let μ : f.Eigenvalues := default
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@ -137,7 +138,9 @@ protected noncomputable def SchurTriangulationAux.of
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by simp
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have hf {bi i' bj j'} (hi : e i = ⟨bi, i'⟩) (hj : e j = ⟨bj, j'⟩) :=
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calc toMatrixOrthonormal basis f i j
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_ = toMatrixOrthonormal bE f (e i) (e j) := by rw [f.toMatrixOrthonormal_reindex] ; rfl
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_ = toMatrixOrthonormal bE f (e i) (e j) := by
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rw [f.toMatrixOrthonormal_reindex]
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rfl
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_ = ⟪bE (e i), f (bE (e j))⟫_𝕜 := f.toMatrixOrthonormal_apply_apply ..
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_ = ⟪(B bi i' : E), f (B bj j')⟫_𝕜 := by rw [hB, hB, hi, hj]
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@ -198,8 +201,8 @@ variable [RCLike 𝕜] [IsAlgClosed 𝕜] [Fintype n] [DecidableEq n] [LinearOrd
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`Matrix.schurTriangulationUnitary`, and `Matrix.schurTriangulation` for which this is their
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simultaneous definition. This is `LinearMap.SchurTriangulationAux` adapted for matrices in the
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Euclidean space. -/
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noncomputable def schurTriangulationAux
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: OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n) × UpperTriangular n 𝕜 :=
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noncomputable def schurTriangulationAux :
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OrthonormalBasis n 𝕜 (EuclideanSpace 𝕜 n) × UpperTriangular n 𝕜 :=
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let f := toEuclideanLin A
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let ⟨d, hd, b, hut⟩ := LinearMap.SchurTriangulationAux.of f
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let e : Fin d ≃o n := Fintype.orderIsoFinOfCardEq n (finrank_euclideanSpace.symm.trans hd)
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@ -208,8 +211,9 @@ noncomputable def schurTriangulationAux
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suffices B.IsUpperTriangular from ⟨b', B, this⟩
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fun i j (hji : j < i) =>
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calc LinearMap.toMatrixOrthonormal b' f i j
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_ = LinearMap.toMatrixOrthonormal b f (e.symm i) (e.symm j) :=
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by rw [f.toMatrixOrthonormal_reindex] ; rfl
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_ = LinearMap.toMatrixOrthonormal b f (e.symm i) (e.symm j) := by
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rw [f.toMatrixOrthonormal_reindex]
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rfl
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_ = 0 := hut (e.symm.lt_iff_lt.mpr hji)
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/-- The change of basis that induces the upper triangular form `A.schurTriangulation` of a matrix
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@ -231,8 +235,8 @@ noncomputable def schurTriangulation : UpperTriangular n 𝕜 :=
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/-- **Schur triangulation**, **Schur decomposition** for matrices over an algebraically closed
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field. In particular, a complex matrix can be converted to upper-triangular form by a change of
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basis. In other words, any complex matrix is unitarily similar to an upper triangular matrix. -/
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theorem schur_triangulation
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: A = A.schurTriangulationUnitary * A.schurTriangulation * star A.schurTriangulationUnitary :=
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lemma schur_triangulation :
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A = A.schurTriangulationUnitary * A.schurTriangulation * star A.schurTriangulationUnitary :=
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let U := A.schurTriangulationUnitary
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have h : U * A.schurTriangulation.val = A * U :=
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let b := A.schurTriangulationBasis.toBasis
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@ -243,6 +247,6 @@ theorem schur_triangulation
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_ = A * U := by simp
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calc A
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_ = A * U * star U := by simp [mul_assoc]
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_ = U * A.schurTriangulation * star U := by rw [←h]
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_ = U * A.schurTriangulation * star U := by rw [← h]
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end Matrix
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