| import Mathlib |
| import OAI.Geometry.Borsuk.MatrixGeometry.SymmetricCoordinates |
| |
| namespace OAI |
|
|
| namespace BorsukNine |
| open scoped Matrix RealInnerProductSpace MatrixOrder Matrix.Norms.L2Operator |
| section SymmetricEntries |
| variable {ν : Type*} [Fintype ν] [LinearOrder ν] |
|
|
| noncomputable def symmetricEntries : Matrix ν ν ℝ →ₗ[ℝ] EuclideanSpace ℝ (UpperTriangle ν) where |
| toFun Q := WithLp.toLp 2 (fun p => Q p.val.1 p.val.2) |
| map_add' _ _ := rfl |
| map_smul' _ _ := rfl |
|
|
| omit [Fintype ν] in |
| lemma symmetricEntries_coordinates (x : EuclideanSpace ℝ (UpperTriangle ν)) : |
| symmetricEntries (symmetricCoordinates x) = x := by |
| ext p |
| exact symmetricCoordinates_upper x _ _ p.property |
|
|
| omit [Fintype ν] in |
| lemma symmetricCoordinates_entries {Q : Matrix ν ν ℝ} (hQ : IsSelfAdjoint Q) : |
| symmetricCoordinates (symmetricEntries Q) = Q := by |
| ext i j |
| change (if h : i ≤ j then Q i j else Q j i) = Q i j |
| split_ifs with h |
| · rfl |
| · have he := congrArg (fun B : Matrix ν ν ℝ => B i j) hQ.star_eq |
| simpa only [Matrix.star_eq_conjTranspose,Matrix.conjTranspose_apply,star_trivial] using he |
|
|
| end SymmetricEntries |
| end BorsukNine |
| |
| namespace BorsukNine |
| open scoped RealInnerProductSpace |
| lemma orthogonal_frame (k : ℕ) {u : EuclideanSpace ℝ (Fin (k+1))} (hu : ‖u‖ = 1) : |
| ∃ e : EuclideanSpace ℝ (Fin k) →ₗᵢ[ℝ] EuclideanSpace ℝ (Fin (k+1)), |
| (∀ x, ⟪u,e x⟫ = 0) ∧ |
| (∀ v, ⟪u,v⟫ = 0 → ∃ x, e x = v) := by |
| let W := (Submodule.span ℝ {u})ᗮ |
| have hu0 : u ≠ 0 := by intro h |
| have hdim : Module.finrank ℝ W = k := by |
| let : Fact (Module.finrank ℝ (EuclideanSpace ℝ (Fin (k+1))) = k+1) := ⟨by simp⟩ |
| exact Submodule.finrank_orthogonal_span_singleton hu0 |
| let b := (stdOrthonormalBasis ℝ W).reindex (finCongr hdim) |
| refine ⟨W.subtypeₗᵢ.comp b.repr.symm.toLinearIsometry,?_,?_⟩ |
| · intro x |
| exact (Submodule.mem_orthogonal_singleton_iff_inner_right.mp (b.repr.symm x).property) |
| · intro v hv |
| have hvW : v ∈ W := Submodule.mem_orthogonal_singleton_iff_inner_right.mpr hv |
| refine ⟨b.repr ⟨v,hvW⟩,?_⟩ |
| change (b.repr.symm (b.repr ⟨v,hvW⟩)).val = v |
| rw [LinearIsometryEquiv.symm_apply_apply] |
| end BorsukNine |
| |
| end OAI |
|
|
