| import Mathlib | |
| import OAI.Geometry.Borsuk.MatrixGeometry.Rank | |
| import OAI.Geometry.Borsuk.MatrixGeometry.PositiveSlice | |
| namespace OAI | |
| namespace BorsukNine | |
| open scoped RealInnerProductSpace Matrix MatrixOrder Matrix.Norms.L2Operator | |
| section MatrixFibre | |
| variable {ν : Type*} [Fintype ν] [DecidableEq ν] | |
| local notation "E" => EuclideanSpace ℝ ν | |
| local notation "M" => Matrix ν ν ℝ | |
| local notation "op" => Matrix.toEuclideanCLM (n := ν) (𝕜 := ℝ) | |
| lemma psd_inner_nonneg {Q : M} (hQ : 0 ≤ Q) (v : E) : 0 ≤ ⟪v,op Q v⟫ := by | |
| have h := (Matrix.nonneg_iff_posSemidef.mp hQ).re_dotProduct_nonneg v.ofLp | |
| simp only [PiLp.inner_apply,RCLike.inner_apply,conj_trivial] | |
| change 0 ≤ ∑ i, (Q *ᵥ v.ofLp) i * v i | |
| simpa only [RCLike.re_to_real,dotProduct,Pi.star_apply,star_trivial,mul_comm] using h | |
| lemma rankOne_mul_of_kernel {Q : M} (hQ : IsSelfAdjoint Q) {u : E} | |
| (hu : op Q u = 0) : rankOne u * Q = 0 := by | |
| apply (Matrix.toEuclideanCLM (n := ν) (𝕜 := ℝ)).injective | |
| change op (rankOne u * Q) = op 0 | |
| rw [map_mul,map_zero] | |
| apply ContinuousLinearMap.ext | |
| intro v | |
| change op (rankOne u) (op Q v) = 0 | |
| rw [rankOne_action] | |
| have hi : ⟪u,op Q v⟫ = 0 := by | |
| rw [← ContinuousLinearMap.adjoint_inner_left, | |
| ← ContinuousLinearMap.star_eq_adjoint,← map_star,hQ.star_eq,hu,inner_zero_left] | |
| rw [hi,zero_smul] | |
| lemma rankOne_split_parts {Q : M} (hQ : 0 ≤ Q) {u : E} (hu : op Q u = 0) | |
| {lam mu : ℝ} (hl : 0 ≤ lam) (hm : 0 ≤ mu) : | |
| (lam • rankOne u - mu • Q)⁺ = lam • rankOne u ∧ | |
| (lam • rankOne u - mu • Q)⁻ = mu • Q := by | |
| apply CFC.posPart_negPart_unique rfl _ (smul_nonneg hl (rankOne_nonneg u)) (smul_nonneg hm hQ) | |
| rw [Matrix.smul_mul,Matrix.mul_smul,rankOne_mul_of_kernel hQ.isSelfAdjoint hu,smul_zero,smul_zero] | |
| lemma nonunit_has_kernel (A : M) (hA : ¬ IsUnit A) : | |
| ∃ v : E, v ≠ 0 ∧ op A v = 0 := by | |
| have hdet : A.det = 0 := by | |
| by_contra h | |
| exact hA ((Matrix.isUnit_iff_isUnit_det A).mpr (isUnit_iff_ne_zero.mpr h)) | |
| obtain ⟨v,hv,hAv⟩ := Matrix.exists_mulVec_eq_zero_iff.mpr hdet | |
| refine ⟨WithLp.toLp 2 v,fun h => hv (WithLp.toLp_injective 2 h),?_⟩ | |
| rw [Matrix.toEuclideanCLM_toLp,hAv,WithLp.toLp_zero] | |
| lemma fibre_boundary_kernel {Q : M} (hQ : 0 ≤ Q) (u : E) | |
| (hnot : ¬ IsUnit (rankOne u + Q)) : | |
| ∃ v : E, ‖v‖ = 1 ∧ ⟪u,v⟫ = 0 ∧ op Q v = 0 := by | |
| obtain ⟨v,hv,hAv⟩ := nonunit_has_kernel _ hnot | |
| have hsum : ⟪v,op (rankOne u + Q) v⟫ = 0 := by rw [hAv,inner_zero_right] | |
| rw [map_add,add_apply,inner_add_right,rankOne_action, | |
| inner_smul_right,real_inner_comm u v] at hsum | |
| have hi : ⟪u,v⟫ = 0 := by | |
| nlinarith [psd_inner_nonneg hQ v,sq_nonneg ⟪u,v⟫] | |
| have hQv : op Q v = 0 := by | |
| simpa only [map_add,add_apply,rankOne_action,hi,zero_smul,zero_add] using hAv | |
| refine ⟨‖v‖⁻¹ • v,by simp [norm_smul,norm_inv,hv],?_,?_⟩ | |
| · simp only [inner_smul_right,hi,mul_zero] | |
| · rw [map_smul,hQv,smul_zero] | |
| noncomputable def matrixFibre (u : E) : Set M := | |
| {Q | 0 ≤ Q ∧ Matrix.trace Q = 1 ∧ op Q u = 0} | |
| lemma matrixFibre_compact (u : E) : IsCompact (matrixFibre u) := by | |
| have hop : Continuous (fun Q : M => op Q u) := | |
| (((op).toAlgEquiv.toLinearEquiv.toLinearMap.continuous_of_finiteDimensional).clm_apply continuous_const) | |
| have he : matrixFibre u = {Q : M | 0 ≤ Q ∧ Matrix.trace Q = 1} ∩ {Q : M | op Q u = 0} := by | |
| ext Q | |
| simp only [matrixFibre,Set.mem_ofPred_eq,Set.mem_inter_iff,and_assoc] | |
| rw [he] | |
| exact positive_slice_compact.inter_right (isClosed_eq hop continuous_const) | |
| end MatrixFibre | |
| end BorsukNine | |
| end OAI | |