| import OAI.Analysis.HyperbolicCones.MatrixComplex |
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| namespace OAI |
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| noncomputable section |
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| open scoped BigOperators Matrix.Norms.L2Operator MatrixOrder ComplexOrder |
| open Matrix |
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| namespace Paper256 |
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| theorem isUnit_of_adjoint_difference_posDef {n : ℕ} (H M : Mat n ℂ) (δ : ℂ) |
| (hδ : δ ≠ 0) (hM : M.PosDef) (hdiff : H - Hᴴ = δ • M) : IsUnit H := by |
| apply Matrix.mulVec_injective_iff_isUnit.mp |
| change Function.Injective H.mulVecLin |
| apply LinearMap.ker_eq_bot.mp |
| apply LinearMap.ker_eq_bot'.2 |
| intro w hw |
| change H *ᵥ w = 0 at hw |
| by_contra hw0 |
| have hq : dotProduct (star w) ((H - Hᴴ) *ᵥ w) = 0 := by |
| rw [Matrix.sub_mulVec, dotProduct_sub, hw, dotProduct_zero] |
| rw [dotProduct_mulVec, vecMul_conjTranspose, star_star, hw, star_zero, |
| zero_dotProduct, sub_self] |
| rw [hdiff, Matrix.smul_mulVec, dotProduct_smul, smul_eq_mul] at hq |
| exact (ne_of_gt (hM.dotProduct_mulVec_pos hw0)) ((mul_eq_zero.mp hq).resolve_left hδ) |
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| theorem resolvent_adjoint_difference {n : ℕ} (A : Mat n ℂ) (hA : A.IsHermitian) |
| (z : ℂ) (hAz : IsUnit (Matrix.det (z • (1 : Mat n ℂ) - A))) : |
| (z • (1 : Mat n ℂ) - A)⁻¹ - ((z • (1 : Mat n ℂ) - A)⁻¹)ᴴ = |
| -(z - star z) • (((z • (1 : Mat n ℂ) - A)⁻¹)ᴴ * |
| (z • (1 : Mat n ℂ) - A)⁻¹) := by |
| let T := z • (1 : Mat n ℂ) - A |
| have ht : T * T⁻¹ = 1 := Matrix.mul_nonsing_inv T hAz |
| have hts : T⁻¹ᴴ * Tᴴ = 1 := by |
| rw [← Matrix.conjTranspose_mul, ht, Matrix.conjTranspose_one] |
| have hd : Tᴴ - T = -(z - star z) • (1 : Mat n ℂ) := by |
| simp only [T, Matrix.conjTranspose_sub, Matrix.conjTranspose_smul, |
| Matrix.conjTranspose_one, hA.eq] |
| module |
| change T⁻¹ - T⁻¹ᴴ = -(z - star z) • (T⁻¹ᴴ * T⁻¹) |
| calc |
| T⁻¹ - T⁻¹ᴴ = T⁻¹ᴴ * (Tᴴ - T) * T⁻¹ := by |
| rw [Matrix.mul_sub, Matrix.sub_mul, hts, Matrix.one_mul, Matrix.mul_assoc, ht, |
| Matrix.mul_one] |
| _ = -(z - star z) • (T⁻¹ᴴ * T⁻¹) := by |
| rw [hd, Matrix.mul_smul, Matrix.mul_one, Matrix.smul_mul] |
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| theorem positive_map_resolvent_isUnit {n m : ℕ} (A : Mat n ℂ) (B : Mat m ℂ) |
| (hA : A.IsHermitian) (hB : B.IsHermitian) |
| (Φ : Mat n ℂ →ₗ[ℂ] Mat m ℂ) |
| (hΦstar : ∀ X, (Φ X)ᴴ = Φ Xᴴ) |
| (hΦpos : ∀ X, X.PosSemidef → (Φ X).PosSemidef) |
| (z : ℂ) (hz : z.im ≠ 0) : |
| IsUnit (z • (1 : Mat m ℂ) - B - Φ ((z • (1 : Mat n ℂ) - A)⁻¹)) := by |
| have hu : IsUnit (z • (1 : Mat n ℂ) - A) := by |
| by_contra hn |
| apply hz |
| apply hA.isSelfAdjoint.im_eq_zero_of_mem_spectrum |
| simpa only [spectrum.mem_iff, Algebra.algebraMap_eq_smul_one] using hn |
| have hd := resolvent_adjoint_difference A hA z ((Matrix.isUnit_iff_isUnit_det _).1 hu) |
| let R := (z • (1 : Mat n ℂ) - A)⁻¹ |
| let M := (1 : Mat m ℂ) + Φ (Rᴴ * R) |
| have hm : M.PosDef := Matrix.PosDef.one.add_posSemidef |
| (hΦpos _ (Matrix.posSemidef_conjTranspose_mul_self R)) |
| refine isUnit_of_adjoint_difference_posDef _ M (z - star z) ?_ hm ?_ |
| · intro he |
| have hi := congrArg Complex.im he |
| simp only [Complex.sub_im, Complex.star_def, Complex.conj_im, Complex.zero_im] at hi |
| apply hz |
| linarith |
| · have hd' : R - Rᴴ = -(z - star z) • (Rᴴ * R) := hd |
| calc |
| _ = (z - star z) • (1 : Mat m ℂ) - Φ (R - Rᴴ) := by |
| simp only [Matrix.conjTranspose_sub, Matrix.conjTranspose_smul, |
| Matrix.conjTranspose_one, hB.eq, hΦstar, map_sub] |
| module |
| _ = (z - star z) • M := by |
| rw [hd', map_smul] |
| dsimp [M] |
| module |
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| end Paper256 |
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| end |
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| end OAI |
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