| import OAI.Geometry.CylinderCovering.Convexity.CompactSpectralBounds |
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| namespace OAI |
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| noncomputable section |
| open Set |
| namespace RuledApproximation |
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| 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 |
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| 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 |
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| 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 |
|
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| end RuledApproximation |
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| end |
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| end OAI |
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