File size: 4,382 Bytes
785c727 | 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 98 99 100 101 | import OAI.Geometry.CylinderCovering.Convexity.CompactSpectralBounds
namespace OAI
noncomputable section
open Set
namespace RuledApproximation
lemma exists_unit_eigenvector (A : Plane →L[ℝ] Plane) {lam : ℝ}
(h : Module.End.HasEigenvalue A.toLinearMap lam) :
∃ v : Plane, ‖v‖ = 1 ∧ Module.End.HasEigenvector A.toLinearMap lam v := by
obtain ⟨u, hu⟩ := h.exists_hasEigenvector
have hu0 : u ≠ 0 := hu.2
have hn : ‖u‖ ≠ 0 := norm_ne_zero_iff.mpr hu0
refine ⟨‖u‖⁻¹ • u, ?_, ?_⟩
· rw [norm_smul, Real.norm_eq_abs, abs_of_nonneg (inv_nonneg.mpr (norm_nonneg _)),
inv_mul_cancel₀ hn]
· refine ⟨Module.End.mem_eigenspace_iff.mpr ?_, smul_ne_zero (inv_ne_zero hn) hu0⟩
rw [map_smul, hu.apply_eq_smul]
module
lemma normalized_eigenframe (A : Plane →L[ℝ] Plane) {lam : ℝ} (hlam : 0 < lam)
(hplus : Module.End.HasEigenvalue A.toLinearMap lam)
(hminus : Module.End.HasEigenvalue A.toLinearMap (-lam)) :
∃ (M : Coordinates ≃L[ℝ] Plane) (v w : Plane),
‖v‖ = 1 ∧ ‖w‖ = 1 ∧ A v = lam • v ∧ A w = (-lam) • w ∧
(∀ z : Coordinates, M z = z 0 • v + z 1 • w) ∧
(∀ z : Coordinates, M.symm (A (M z)) = ![lam * z 0, -lam * z 1]) ∧
‖M.toContinuousLinearMap‖ ≤ 2 := by
obtain ⟨v, hvn, hv⟩ := exists_unit_eigenvector A hplus
obtain ⟨w, hwn, hw⟩ := exists_unit_eigenvector A hminus
have hvals : Function.Injective (![lam, -lam] : Fin 2 → ℝ) := by
intro i j hij
fin_cases i <;> fin_cases j <;> simp at hij ⊢ <;> linarith
have hli : LinearIndependent ℝ (![v, w] : Fin 2 → Plane) :=
Module.End.eigenvectors_linearIndependent' A.toLinearMap ![lam, -lam] hvals ![v, w] (by
intro i
fin_cases i
· exact hv
· exact hw)
let B : Module.Basis (Fin 2) ℝ Plane :=
basisOfLinearIndependentOfCardEqFinrank hli (by simp)
let M : Coordinates ≃L[ℝ] Plane := B.equivFun.symm.toContinuousLinearEquiv
have hM (z : Coordinates) : M z = z 0 • v + z 1 • w := by
simp [M, B]
have hAv : A v = lam • v := hv.apply_eq_smul
have hAw : A w = (-lam) • w := hw.apply_eq_smul
refine ⟨M, v, w, hvn, hwn, hAv, hAw, hM, ?_, ?_⟩
· intro z
apply M.injective
rw [M.apply_symm_apply, hM, map_add, map_smul, map_smul,
hAv, hAw, hM]
simp only [Matrix.cons_val_zero, Matrix.cons_val_one]
module
· apply ContinuousLinearMap.opNorm_le_bound _ (by norm_num)
intro z
rw [ContinuousLinearEquiv.coe_coe, hM]
calc
‖z 0 • v + z 1 • w‖ ≤ ‖z 0 • v‖ + ‖z 1 • w‖ := norm_add_le _ _
_ = ‖z 0‖ + ‖z 1‖ := by rw [norm_smul, norm_smul, hvn, hwn]; ring
_ ≤ 2 * ‖z‖ := by
have h0 := norm_le_pi_norm z (0 : Fin 2)
have h1 := norm_le_pi_norm z (1 : Fin 2)
linarith
lemma eigenframe_inverse_bound_of_coefficients
(A : Plane →L[ℝ] Plane) (M : Coordinates ≃L[ℝ] Plane) (v w : Plane)
{m B lam : ℝ} (hm : 0 < m) (hml : m ≤ lam) (hlB : lam ≤ B)
(hAB : ‖A‖ ≤ B) (hM : ∀ z : Coordinates, M z = z 0 • v + z 1 • w)
(hcoeff : ∀ a b : ℝ, lam * (|a| + |b|) ≤
(‖A‖ + lam) * ‖a • v + b • w‖) :
‖M.symm.toContinuousLinearMap‖ ≤ 2 * B / m := by
have hB : 0 ≤ B := (hm.le.trans hml).trans hlB
apply ContinuousLinearMap.opNorm_le_bound _ (by positivity)
intro p
let z : Coordinates := M.symm p
have hz : ‖z‖ ≤ |z 0| + |z 1| := by
apply (pi_norm_le_iff_of_nonneg (by positivity)).mpr
intro i
fin_cases i <;> simp only [Real.norm_eq_abs]
· exact le_add_of_nonneg_right (abs_nonneg _)
· exact le_add_of_nonneg_left (abs_nonneg _)
have hc : lam * (|z 0| + |z 1|) ≤ (‖A‖ + lam) * ‖p‖ := by
have h := hcoeff (z 0) (z 1)
rw [← hM, show M z = p from M.apply_symm_apply p] at h
exact h
have hbound : m * ‖z‖ ≤ (2 * B) * ‖p‖ := calc
m * ‖z‖ ≤ m * (|z 0| + |z 1|) := mul_le_mul_of_nonneg_left hz hm.le
_ ≤ lam * (|z 0| + |z 1|) := mul_le_mul_of_nonneg_right hml (by positivity)
_ ≤ (‖A‖ + lam) * ‖p‖ := hc
_ ≤ (2 * B) * ‖p‖ := mul_le_mul_of_nonneg_right (by linarith) (norm_nonneg _)
change ‖z‖ ≤ 2 * B / m * ‖p‖
rw [div_mul_eq_mul_div, le_div_iff₀ hm]
nlinarith
end RuledApproximation
end
end OAI
|