| import Mathlib |
| import OAI.Geometry.Borsuk.MatrixGeometry.SignedExtension |
|
|
| namespace OAI |
|
|
| namespace BorsukNine |
| open scoped RealInnerProductSpace ContDiff |
| open scoped Manifold |
| open MeasureTheory Set Metric |
| open scoped ENNReal Pointwise Matrix |
| section FullSupport |
| 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) |
|
|
| lemma Admissible.positiveExtension_kernel {f : E → ι → ℝ} (hf : Admissible f) |
| {Q : Matrix ν ν ℝ} (hQ : 0 ≤ Q) {w : E} (hw : ‖w‖ = 1) |
| (hk : op Q w = 0) {i : ι} (hi : 0 < f w i) : positiveExtension f Q i = 0 := by |
| apply le_antisymm _ (hf.positiveExtension_nonneg Q i) |
| apply le_of_not_gt |
| intro hp |
| obtain ⟨v, ⟨x, rfl⟩, hv, hx⟩ := (hf.positiveExtension_support hQ i).mp hp |
| exact hf.orthogonal _ _ hw hv (matrix_image_orthogonal_kernel hQ.isSelfAdjoint hk x) i |
| ⟨hi, hx⟩ |
|
|
| lemma Admissible.signedExtension_kernel {f : E → ι → ℝ} (hf : Admissible f) |
| {A : Matrix ν ν ℝ} (hA : IsSelfAdjoint A) {w : E} (hw : ‖w‖ = 1) |
| (hk : op A w = 0) : ∃ i, signedExtension f A i = 0 := by |
| obtain ⟨i, hi⟩ := hf.support_nonempty hw |
| have hi' := (mem_positiveSupport f w i).mp hi |
| obtain ⟨hp, hn⟩ := matrix_parts_kernel hA hk |
| refine ⟨i, ?_⟩ |
| simp only [signedExtension, Pi.sub_apply, |
| hf.positiveExtension_kernel (CFC.posPart_nonneg A) hw hp hi', |
| hf.positiveExtension_kernel (CFC.negPart_nonneg A) hw hn hi', sub_self] |
|
|
| lemma Admissible.signedExtension_kernel_eq_zero {f : E → ι → ℝ} (hf : Admissible f) |
| {A : Matrix ν ν ℝ} (hA : IsSelfAdjoint A) (hs : ∀ i, signedExtension f A i ≠ 0) |
| {w : E} (hk : op A w = 0) : w = 0 := by |
| by_contra hw |
| have hunit : ‖‖w‖⁻¹ • w‖ = 1 := by simp [norm_smul, norm_inv, hw] |
| have hk' : op A (‖w‖⁻¹ • w) = 0 := by rw [map_smul, hk, smul_zero] |
| obtain ⟨i, hi⟩ := hf.signedExtension_kernel hA hunit hk' |
| exact hs i hi |
|
|
| lemma Admissible.signedExtension_fullSupport {f : E → ι → ℝ} (hf : Admissible f) |
| {A : Matrix ν ν ℝ} (hA : IsSelfAdjoint A) (hs : ∀ i, signedExtension f A i ≠ 0) : |
| IsUnit A := by |
| apply Matrix.mulVec_injective_iff_isUnit.mp |
| intro u v huv |
| have hz : op A (WithLp.toLp 2 u - WithLp.toLp 2 v) = 0 := by |
| rw [map_sub, Matrix.toEuclideanCLM_toLp, Matrix.toEuclideanCLM_toLp, huv, sub_self] |
| have h := sub_eq_zero.mp (hf.signedExtension_kernel_eq_zero hA hs hz) |
| exact WithLp.toLp_injective 2 h |
|
|
| omit [Nonempty ν] in |
| lemma matrix_psd_trace_zero {Q : Matrix ν ν ℝ} (hQ : 0 ≤ Q) |
| (htrace : Matrix.trace Q = 0) : Q = 0 := by |
| have hprod : (CFC.sqrt Q)ᴴ * CFC.sqrt Q = Q := by |
| rw [← Matrix.star_eq_conjTranspose, (CFC.sqrt_nonneg Q).star_eq, CFC.sqrt_mul_sqrt_self Q hQ] |
| have hz : CFC.sqrt Q = 0 := Matrix.trace_conjTranspose_mul_self_eq_zero_iff.mp |
| (hprod.symm ▸ htrace) |
| rw [← hprod, hz] |
| simp |
|
|
| lemma Admissible.signedExtension_negativeDefinite {f : E → ι → ℝ} (hf : Admissible f) |
| {A : Matrix ν ν ℝ} (hA : IsSelfAdjoint A) (hs : ∀ i, signedExtension f A i < 0) : |
| (-A).PosDef := by |
| have hp (i : ι) : positiveExtension f A⁺ i = 0 := by |
| have h := hf.positiveExtension_orthogonal (CFC.posPart_nonneg A) |
| (CFC.negPart_nonneg A) (CFC.posPart_mul_negPart A) i |
| rcases mul_eq_zero.mp h with h | h |
| · exact h |
| · have hi := hs i |
| simp only [signedExtension, Pi.sub_apply, h, sub_zero] at hi |
| exact False.elim ((not_lt_of_ge (hf.positiveExtension_nonneg A⁺ i)) hi) |
| have hzero : A⁺ = 0 := matrix_psd_trace_zero (CFC.posPart_nonneg A) (by |
| rw [← hf.positiveExtension_sum (CFC.posPart_nonneg A)] |
| simp [hp]) |
| have hneg : -A = A⁻ := by |
| have h := CFC.posPart_sub_negPart A hA |
| rw [hzero, zero_sub] at h |
| exact (congrArg Neg.neg h).symm.trans (neg_neg A⁻) |
| have hn : 0 ≤ -A := hneg ▸ CFC.negPart_nonneg A |
| apply (Matrix.nonneg_iff_posSemidef.mp hn).posDef_iff_isUnit.mpr |
| exact (hf.signedExtension_fullSupport hA (fun i => ne_of_lt (hs i))).neg |
|
|
| end FullSupport |
|
|
| end BorsukNine |
|
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| end OAI |
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