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9eeb9c7 | 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 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 | import OAI.Analysis.HyperbolicCones.MatrixPositive
namespace OAI
/-! Spectral identities for the inverse square root. -/
noncomputable section
open scoped Matrix.Norms.L2Operator MatrixOrder
open Matrix
namespace Paper256
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]
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)
theorem inverseSquareRoot_isHermitian {n : ℕ} (H : Sym n) :
(inverseSquareRoot H).IsHermitian := by
rw [inverseSquareRoot_eq_cfc]
exact IsSelfAdjoint.cfc
theorem inverseSquareRoot_transpose {n : ℕ} (H : Sym n) :
(inverseSquareRoot H)ᵀ = inverseSquareRoot H :=
Matrix.isHermitian_iff_isSymm.mp (inverseSquareRoot_isHermitian H)
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
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)
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
end Paper256
end
end OAI
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