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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 92 93 94 95 96 97 98 99 100 | import Mathlib
import OAI.Geometry.Borsuk.SphereGeometry.SphereMoments
import OAI.Geometry.Borsuk.SphereGeometry.IntegralLift
import OAI.Geometry.Borsuk.MatrixGeometry.MatrixNorm
namespace OAI
namespace BorsukNine
open scoped RealInnerProductSpace ContDiff
open scoped Manifold
open MeasureTheory Set Metric
open scoped ENNReal Pointwise Matrix
section MatrixKernel
open scoped MatrixOrder Matrix.Norms.L2Operator
variable {ν : Type*} [Fintype ν] [DecidableEq ν]
local notation "Ek" => EuclideanSpace ℝ ν
local notation "op" => Matrix.toEuclideanCLM (n := ν) (𝕜 := ℝ)
lemma matrix_abs_action_norm_sq (A : Matrix ν ν ℝ) (x : Ek) :
‖op (CFC.abs A) x‖ ^ 2 = ‖op A x‖ ^ 2 := by
rw [matrix_action_norm_sq, (CFC.abs_nonneg A).star_eq, CFC.abs_mul_abs,
← matrix_action_norm_sq]
lemma matrix_abs_kernel {A : Matrix ν ν ℝ} {x : Ek} (hx : op A x = 0) :
op (CFC.abs A) x = 0 := by
apply norm_eq_zero.mp
apply sq_eq_zero_iff.mp
rw [matrix_abs_action_norm_sq, hx, norm_zero, zero_pow (by decide : 2 ≠ 0)]
lemma matrix_parts_kernel {A : Matrix ν ν ℝ} (hA : IsSelfAdjoint A)
{x : Ek} (hx : op A x = 0) : op A⁺ x = 0 ∧ op A⁻ x = 0 := by
have habs := matrix_abs_kernel hx
constructor
· have h := congrArg (fun B : Matrix ν ν ℝ => op B x) (CFC.abs_add_self A hA)
have hh : (2 : ℝ) • op A⁺ x = 0 := by
simpa only [map_add, add_apply, two_smul, hx, habs, add_zero] using h.symm
exact (smul_eq_zero.mp hh).resolve_left (by norm_num)
· have h := congrArg (fun B : Matrix ν ν ℝ => op B x) (CFC.abs_sub_self A hA)
have hh : (2 : ℝ) • op A⁻ x = 0 := by
simpa only [map_sub, sub_apply, two_smul, hx, habs, sub_zero,
map_add, add_apply] using h.symm
exact (smul_eq_zero.mp hh).resolve_left (by norm_num)
lemma matrix_image_orthogonal_kernel {A : Matrix ν ν ℝ} (hA : IsSelfAdjoint A)
{x : Ek} (hx : op A x = 0) (y : Ek) : ⟪x, op A y⟫ = 0 := by
rw [← ContinuousLinearMap.adjoint_inner_left, ← ContinuousLinearMap.star_eq_adjoint,
← map_star, hA.star_eq, hx, inner_zero_left]
end MatrixKernel
section PositiveExtension
open scoped MatrixOrder Matrix.Norms.L2Operator
variable {ν ι : Type*} [Fintype ν] [DecidableEq ν] [Nonempty ν] [Fintype ι]
local notation "E" => EuclideanSpace ℝ ν
local notation "op" => Matrix.toEuclideanCLM (n := ν) (𝕜 := ℝ)
noncomputable def positiveExtension (f : E → ι → ℝ) (Q : Matrix ν ν ℝ) : ι → ℝ :=
integralLift f (op (CFC.sqrt Q))
omit [Nonempty ν] in
lemma Admissible.positiveExtension_nonneg {f : E → ι → ℝ} (hf : Admissible f)
(Q : Matrix ν ν ℝ) (i : ι) : 0 ≤ positiveExtension f Q i :=
hf.integralLift_nonneg _ _
lemma Admissible.positiveExtension_sum {f : E → ι → ℝ} (hf : Admissible f)
{Q : Matrix ν ν ℝ} (hQ : 0 ≤ Q) : ∑ i, positiveExtension f Q i = Matrix.trace Q := by
classical
rw [positiveExtension, hf.integralLift_sum]
simp_rw [sqrt_action_norm_sq Q hQ, Matrix.inner_toEuclideanCLM]
rw [sphere_quadratic_integral]
have hn : (Fintype.card ν : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr (ne_of_gt Fintype.card_pos)
field_simp
lemma Admissible.positiveExtension_continuousOn {f : E → ι → ℝ} (hf : Admissible f) :
ContinuousOn (positiveExtension f) {Q : Matrix ν ν ℝ | 0 ≤ Q} := by
have hc : Continuous (fun Q : Matrix ν ν ℝ => op Q) :=
(op).toAlgEquiv.toLinearEquiv.toLinearMap.continuous_of_finiteDimensional
exact hf.integralLift_continuous.comp_continuousOn
(hc.comp_continuousOn CFC.continuousOn_sqrt)
omit [Nonempty ν] [Fintype ι] in
lemma positiveExtension_smul_nonneg (f : E → ι → ℝ)
{Q : Matrix ν ν ℝ} (hQ : 0 ≤ Q) {t : ℝ} (ht : 0 ≤ t) :
positiveExtension f (t • Q) = t • positiveExtension f Q := by
unfold positiveExtension
rw [matrix_sqrt_smul hQ ht, map_smul,
integralLift_smul_nonneg f _ (Real.sqrt_nonneg t), Real.sq_sqrt ht]
lemma Admissible.positiveExtension_support {f : E → ι → ℝ} (hf : Admissible f)
{Q : Matrix ν ν ℝ} (hQ : 0 ≤ Q) (i : ι) :
0 < positiveExtension f Q i ↔ ∃ v : E,
v ∈ LinearMap.range (op Q).toLinearMap ∧ ‖v‖ = 1 ∧ 0 < f v i := by
rw [positiveExtension, hf.integralLift_support, matrix_sqrt_range Q hQ]
end PositiveExtension
end BorsukNine
end OAI
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