File size: 3,348 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 | import OAI.Geometry.CylinderCovering.Ruled.LocalDefinitions
import OAI.Geometry.CylinderCovering.Convexity.CompactEigenframes
import OAI.Geometry.CylinderCovering.Convexity.EigenframeBounds
namespace OAI
noncomputable section
open Set
namespace RuledApproximation
lemma positive_eigenvalue_eq_sqrt (A : Plane →L[ℝ] Plane)
(htr : LinearMap.trace ℝ Plane A.toLinearMap = 0)
(hdet : LinearMap.det A.toLinearMap < 0) {lam : ℝ} (hlam : 0 < lam)
(heig : Module.End.HasEigenvalue A.toLinearMap lam) :
lam = Real.sqrt (-LinearMap.det A.toLinearMap) := by
rw [Module.End.hasEigenvalue_iff_isRoot_charpoly, Polynomial.IsRoot,
plane_charpoly, htr] at heig
simp only [Polynomial.eval_add, Polynomial.eval_sub, Polynomial.eval_pow,
Polynomial.eval_X, Polynomial.eval_mul, Polynomial.eval_C, zero_mul, sub_zero] at heig
have hs := Real.sq_sqrt (show 0 ≤ -LinearMap.det A.toLinearMap by linarith)
have hn := Real.sqrt_nonneg (-LinearMap.det A.toLinearMap)
nlinarith
end RuledApproximation
namespace CylinderCovering.ProductLocal
open RuledApproximation
theorem exists_unit_eigenframe (A : Plane →L[ℝ] Plane)
(htr : LinearMap.trace ℝ Plane A.toLinearMap = 0)
(hdet : LinearMap.det A.toLinearMap < 0) :
∃ L : (ℝ × ℝ) ≃L[ℝ] Plane,
IsUnitEigenframe A (Real.sqrt (-LinearMap.det A.toLinearMap)) L := by
have hlam : 0 < Real.sqrt (-LinearMap.det A.toLinearMap) :=
Real.sqrt_pos.mpr (neg_pos.mpr hdet)
obtain ⟨hp, hm⟩ := plane_hyperbolic_eigenvalues A.toLinearMap htr hdet
obtain ⟨M, v, w, hv, hw, hAv, hAw, hM, _, _⟩ := normalized_eigenframe A hlam hp hm
let L : (ℝ × ℝ) ≃L[ℝ] Plane := (ContinuousLinearEquiv.finTwoArrow ℝ ℝ).symm.trans M
have hL0 : L (1,0) = v := by simp [L, hM]
have hL1 : L (0,1) = w := by simp [L, hM]
refine ⟨L, ?_⟩
simp only [IsUnitEigenframe, hL0, hL1]
exact ⟨hv, hw, hAv, hAw⟩
theorem compact_uniform_eigenframe_inverse {X : Type*} [TopologicalSpace X]
{D : Set X} (hD : IsCompact D) {A : X → Plane →L[ℝ] Plane}
(hA : ContinuousOn A D)
(htr : ∀ p ∈ D, LinearMap.trace ℝ Plane (A p).toLinearMap = 0)
(hdet : ∀ p ∈ D, LinearMap.det (A p).toLinearMap < 0) :
∃ N : ℝ, 0 < N ∧ ∀ p ∈ D, ∀ lam : ℝ, ∀ L : (ℝ × ℝ) ≃L[ℝ] Plane,
0 < lam → IsUnitEigenframe (A p) lam L →
‖L.symm.toContinuousLinearMap‖ ≤ N ∧ ∀ z : Plane, ‖L.symm z‖ ≤ N * ‖z‖ := by
obtain ⟨m, B, hm, hB, hbounds⟩ := compact_spectral_envelope hD hA htr hdet
refine ⟨2*B/m, by positivity, ?_⟩
intro p hp lam L hlam hL
have hv0 : L (1,0) ≠ 0 := by
intro hz
have hn := hL.1
rw [hz, norm_zero] at hn
norm_num at hn
have heig : Module.End.HasEigenvalue (A p).toLinearMap lam :=
Module.End.hasEigenvalue_of_hasEigenvector
⟨Module.End.mem_eigenspace_iff.mpr hL.2.2.1, hv0⟩
have heq := positive_eigenvalue_eq_sqrt (A p) (htr p hp) (hdet p hp) hlam heig
have hml : m ≤ lam := by rw [heq]; exact (hbounds p hp).1
refine ⟨EigenframeBounds.pair_frame_inverse_opNorm (A p) L hm hml
(hbounds p hp).2.2 hL.1 hL.2.1 hL.2.2.1 hL.2.2.2, ?_⟩
exact EigenframeBounds.pair_frame_inverse_bound (A p) L hm hml
(hbounds p hp).2.2 hL.1 hL.2.1 hL.2.2.1 hL.2.2.2
end CylinderCovering.ProductLocal
end
end OAI
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