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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 83 84 85 86 87 88 89 90 91 | import Mathlib
import OAI.Geometry.Borsuk.Support.NegativeFibre
import OAI.Geometry.Borsuk.MatrixGeometry.BlockParts
namespace OAI
open Set MeasureTheory
namespace BorsukNine
open scoped Matrix MatrixOrder RealInnerProductSpace Matrix.Norms.L2Operator
section BlockLine
variable {μ ν : Type*} [Fintype μ] [Fintype ν] [DecidableEq μ] [DecidableEq ν]
[Nonempty ν]
local notation "W" => EuclideanSpace ℝ μ
local notation "E" => EuclideanSpace ℝ ν
local notation "M" => Matrix ν ν ℝ
local notation "op" => Matrix.toEuclideanCLM (𝕜 := ℝ)
lemma positiveLine_eigenvector {A : M} (hA : IsSelfAdjoint A) {c : E} (hc : ‖c‖ = 1)
{lam : ℝ} (hl : 0 < lam) (hP : A⁺ = lam • rankOne c) : op A c = lam • c := by
rw [← CFC.posPart_sub_negPart A hA,map_sub,sub_apply,
negPart_kernel_of_positiveLine hc hl hP,sub_zero,hP,map_smul,smul_apply,
rankOne_action,real_inner_self_eq_norm_sq,hc,one_pow,one_smul]
lemma blockScalar_positiveLine {A : M} (hA : IsSelfAdjoint A) {c : E} (hc : ‖c‖ = 1)
{lam : ℝ} (hl : 0 < lam) (hP : A⁺ = lam • rankOne c) : blockScalar c A = lam := by
rw [blockScalar,positiveLine_eigenvector hA hc hl hP,inner_smul_right,real_inner_self_eq_norm_sq,hc,one_pow,mul_one]
omit [DecidableEq μ] in
lemma blockOff_positiveLine (e : W →ₗᵢ[ℝ] E) {A : M} (hA : IsSelfAdjoint A) {c : E} (hc : ‖c‖ = 1)
(he : ∀ v, ⟪c,e v⟫ = 0) {lam : ℝ} (hl : 0 < lam) (hP : A⁺ = lam • rankOne c) :
blockOff c e A = 0 := by
rw [blockOff,positiveLine_eigenvector hA hc hl hP,map_smul,adjoint_normal c e he,smul_zero]
lemma blockCorner_positiveLine (e : W →ₗᵢ[ℝ] E) {A : M} (hA : IsSelfAdjoint A) {c : E} (hc : ‖c‖ = 1)
(he : ∀ v, ⟪c,e v⟫ = 0) (hes : ∀ u, ⟪c,u⟫ = 0 → ∃ v, e v = u)
{lam : ℝ} (hl : 0 < lam) (hP : A⁺ = lam • rankOne c) : blockCorner e A = -A⁻ := by
have hd := block_decomposition e hc he hes hA (blockOff_positiveLine e hA hc he hl hP)
rw [blockScalar_positiveLine hA hc hl hP] at hd
have hp := CFC.posPart_sub_negPart A hA
rw [hP] at hp
exact (add_left_cancel (show lam • rankOne c + -A⁻ = lam • rankOne c + blockCorner e A by simpa only [sub_eq_add_neg] using hp.trans hd)).symm
omit [Nonempty ν] in
lemma blockCompress_isUnit (e : W →ₗᵢ[ℝ] E) {A : M} (hA : IsSelfAdjoint A) (hU : IsUnit A)
{c : E} (hc : ‖c‖ = 1) (he : ∀ v, ⟪c,e v⟫ = 0)
(hes : ∀ u, ⟪c,u⟫ = 0 → ∃ v, e v = u) (hb : blockOff c e A = 0) :
IsUnit (matrixCompress e A) := by
have hopU : IsUnit (op A) := IsUnit.map (Matrix.toEuclideanCLM (n := ν) (𝕜 := ℝ)) hU
have hAi : Function.Injective (op A) := (ContinuousLinearMap.isUnit_iff_bijective.mp hopU).1
apply (isUnit_map_iff (op) _).mp
apply ContinuousLinearMap.isUnit_iff_bijective.mpr
have hi : Function.Injective (op (matrixCompress e A)) := by
apply (LinearMap.ker_eq_bot).mp
rw [Submodule.eq_bot_iff]
intro v hv
have hv0 : op (matrixCompress e A) v = 0 := hv
have hAv : op A (e v) = 0 := by
conv_lhs => rw [block_decomposition e hc he hes hA hb]
rw [map_add,add_apply,map_smul,smul_apply,
rankOne_action,he,zero_smul,smul_zero,zero_add]
change op (matrixEmbed e (matrixCompress e A)) (e v) = 0
rw [show op (matrixEmbed e (matrixCompress e A)) (e v) = e (op (matrixCompress e A) v) from matrixEmbed_action e _ v,hv0,map_zero]
apply e.injective
exact hAi (by simpa only [map_zero] using hAv)
exact ⟨hi,(LinearMap.injective_iff_surjective).mp hi⟩
lemma posDef_inner_pos {B : Matrix μ μ ℝ} (hB : B.PosDef) {v : W} (hv : v ≠ 0) :
0 < ⟪v,op B v⟫ := by
have hv' : v.ofLp ≠ 0 := fun h => hv (WithLp.ofLp_injective 2 h)
have h := hB.dotProduct_mulVec_pos hv'
simp only [PiLp.inner_apply,RCLike.inner_apply,conj_trivial]
change 0 < ∑ i, (B *ᵥ v.ofLp) i * v i
simpa only [dotProduct,Pi.star_apply,star_trivial,mul_comm] using h
lemma blockCompress_negative (e : W →ₗᵢ[ℝ] E) {A : M} (hA : IsSelfAdjoint A) (hU : IsUnit A)
{c : E} (hc : ‖c‖ = 1) (he : ∀ v, ⟪c,e v⟫ = 0)
(hes : ∀ u, ⟪c,u⟫ = 0 → ∃ v, e v = u) {lam : ℝ} (hl : 0 < lam)
(hP : A⁺ = lam • rankOne c) : NegativeQuadratic (op (matrixCompress e A)) := by
have hC := blockCorner_positiveLine e hA hc he hes hl hP
have hcomp : -(matrixCompress e A) = matrixCompress e A⁻ := by
have hh := congrArg (matrixCompress e) hC
rw [blockCorner,matrixCompress_embed,map_neg] at hh
exact neg_eq_iff_eq_neg.mpr hh
have hn : 0 ≤ -(matrixCompress e A) := by rw [hcomp]; exact matrixCompress_nonneg e (CFC.negPart_nonneg A)
have hunit := (blockCompress_isUnit e hA hU hc he hes (blockOff_positiveLine e hA hc he hl hP)).neg
have hpd := (Matrix.nonneg_iff_posSemidef.mp hn).posDef_iff_isUnit.mpr hunit
intro v hv
have h := posDef_inner_pos hpd hv
rw [map_neg,neg_apply,inner_neg_right] at h
exact neg_pos.mp h
end BlockLine
end BorsukNine
end OAI
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