openAI-math-dataset / lean /OAI /Geometry /CylinderCovering /Convexity /CompactProductEigenframes.lean
| 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 | |