| import OAI.Analysis.HyperbolicCones.MatrixPositive |
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| namespace OAI |
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| /-! Spectral identities for the inverse square root. -/ |
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| noncomputable section |
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| open scoped Matrix.Norms.L2Operator MatrixOrder |
| open Matrix |
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| namespace Paper256 |
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| theorem inverseSquareRoot_eq_cfc {n : ℕ} (H : Sym n) : |
| inverseSquareRoot H = cfc (fun x : ℝ => (Real.sqrt x)⁻¹) (H : Mat n ℝ) := by |
| let hH : (H : Mat n ℝ).IsHermitian := H.property |
| rw [hH.cfc_eq] |
| simp [inverseSquareRoot, Matrix.IsHermitian.cfc, Unitary.conjStarAlgAut_apply, |
| Function.comp_def, Matrix.star_eq_conjTranspose] |
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|
| theorem matrix_cfc_mul {n : ℕ} (H : Sym n) (f g : ℝ → ℝ) : |
| cfc (fun x => f x * g x) (H : Mat n ℝ) = |
| cfc f (H : Mat n ℝ) * cfc g (H : Mat n ℝ) := |
| cfc_mul f g _ (H.val.finite_real_spectrum.continuousOn f) |
| (H.val.finite_real_spectrum.continuousOn g) |
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| theorem inverseSquareRoot_isHermitian {n : ℕ} (H : Sym n) : |
| (inverseSquareRoot H).IsHermitian := by |
| rw [inverseSquareRoot_eq_cfc] |
| exact IsSelfAdjoint.cfc |
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| theorem inverseSquareRoot_transpose {n : ℕ} (H : Sym n) : |
| (inverseSquareRoot H)ᵀ = inverseSquareRoot H := |
| Matrix.isHermitian_iff_isSymm.mp (inverseSquareRoot_isHermitian H) |
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|
| theorem posDef_real_spectrum_positive {n : ℕ} {A : Mat n ℝ} (hA : A.PosDef) |
| {x : ℝ} (hx : x ∈ spectrum ℝ A) : 0 < x := by |
| rw [hA.isHermitian.spectrum_real_eq_range_eigenvalues] at hx |
| obtain ⟨i, rfl⟩ := hx |
| exact hA.eigenvalues_pos i |
|
|
| theorem inverseSquareRoot_conjugate {n : ℕ} (H : Sym n) (hH : (H : Mat n ℝ).PosDef) : |
| (inverseSquareRoot H)ᵀ * (H : Mat n ℝ) * inverseSquareRoot H = 1 := by |
| rw [inverseSquareRoot_transpose, inverseSquareRoot_eq_cfc] |
| let f : ℝ → ℝ := fun x => (Real.sqrt x)⁻¹ |
| have hid : cfc (fun x : ℝ => x) (H : Mat n ℝ) = (H : Mat n ℝ) := |
| cfc_id' ℝ (H : Mat n ℝ) H.property |
| change cfc f (H : Mat n ℝ) * (H : Mat n ℝ) * cfc f (H : Mat n ℝ) = 1 |
| calc |
| cfc f (H : Mat n ℝ) * (H : Mat n ℝ) * cfc f (H : Mat n ℝ) = |
| cfc (fun x => f x * x * f x) (H : Mat n ℝ) := by |
| rw [matrix_cfc_mul H (fun x => f x * x) f, |
| matrix_cfc_mul H f (fun x => x)] |
| rw [hid] |
| _ = cfc (fun _ : ℝ => 1) (H : Mat n ℝ) := by |
| apply cfc_congr |
| intro x hx |
| have hxpos := posDef_real_spectrum_positive hH hx |
| have hs : Real.sqrt x ≠ 0 := (Real.sqrt_pos.mpr hxpos).ne' |
| dsimp [f] |
| field_simp |
| exact (Real.sq_sqrt hxpos.le).symm |
| _ = 1 := cfc_const_one ℝ (H : Mat n ℝ) H.property |
|
|
| theorem inverseSquareRoot_isUnit {n : ℕ} (H : Sym n) (hH : (H : Mat n ℝ).PosDef) : |
| IsUnit (inverseSquareRoot H) := by |
| rw [Matrix.isUnit_iff_isUnit_det, isUnit_iff_ne_zero] |
| intro hs |
| have h := congrArg Matrix.det (inverseSquareRoot_conjugate H hH) |
| simp [Matrix.det_mul, Matrix.det_transpose, hs] at h |
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|
| theorem inverseSquareRoot_commute {n : ℕ} (H : Sym n) : |
| Commute (inverseSquareRoot H) (H : Mat n ℝ) := by |
| rw [inverseSquareRoot_eq_cfc] |
| have hid : cfc (fun x : ℝ => x) (H : Mat n ℝ) = (H : Mat n ℝ) := |
| cfc_id' ℝ (H : Mat n ℝ) H.property |
| simpa only [hid] using |
| cfc_commute_cfc (fun x : ℝ => (Real.sqrt x)⁻¹) (fun x => x) (H : Mat n ℝ) |
|
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| theorem inverseSquareRoot_sq {n : ℕ} (H : Sym n) (hH : (H : Mat n ℝ).PosDef) : |
| inverseSquareRoot H * inverseSquareRoot H = (H : Mat n ℝ)⁻¹ := by |
| have h := inverseSquareRoot_conjugate H hH |
| rw [inverseSquareRoot_transpose] at h |
| have hi : (inverseSquareRoot H * inverseSquareRoot H) * (H : Mat n ℝ) = 1 := by |
| calc |
| _ = inverseSquareRoot H * (inverseSquareRoot H * (H : Mat n ℝ)) := mul_assoc _ _ _ |
| _ = inverseSquareRoot H * ((H : Mat n ℝ) * inverseSquareRoot H) := by |
| rw [(inverseSquareRoot_commute H).eq] |
| _ = 1 := by simpa only [mul_assoc] using h |
| exact (Matrix.inv_eq_left_inv hi).symm |
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| end Paper256 |
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| end |
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| end OAI |
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|
