File size: 3,624 Bytes
8e4d533 | 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 77 78 79 80 81 82 | 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
|