| import Mathlib |
|
|
| namespace OAI |
|
|
| namespace BorsukNine |
| open scoped Matrix MatrixOrder Matrix.Norms.L2Operator |
| section PositiveSlice |
| variable {ν : Type*} [Fintype ν] [DecidableEq ν] |
| local notation "M" => Matrix ν ν ℝ |
| |
| noncomputable def unflattenMatrix : EuclideanSpace ℝ (ν × ν) →ₗ[ℝ] M where |
| toFun x i j := x (i,j) |
| map_add' _ _ := rfl |
| map_smul' _ _ := rfl |
|
|
| noncomputable def flatGram (x : EuclideanSpace ℝ (ν × ν)) : M := |
| unflattenMatrix x * (unflattenMatrix x)ᴴ |
|
|
| omit [DecidableEq ν] in |
| lemma flatGram_nonneg (x : EuclideanSpace ℝ (ν × ν)) : 0 ≤ flatGram x := |
| (Matrix.posSemidef_self_mul_conjTranspose _).nonneg |
|
|
| omit [DecidableEq ν] in |
| lemma flatGram_trace (x : EuclideanSpace ℝ (ν × ν)) : Matrix.trace (flatGram x) = ‖x‖ ^ 2 := by |
| rw [EuclideanSpace.real_norm_sq_eq] |
| simp only [flatGram,Matrix.trace,Matrix.diag,Matrix.mul_apply,Matrix.conjTranspose_apply, |
| star_trivial,Fintype.sum_prod_type] |
| simp only [pow_two] |
| rfl |
|
|
| omit [DecidableEq ν] in |
| lemma flatGram_continuous : Continuous (flatGram (ν := ν)) := by |
| have h := (unflattenMatrix (ν := ν)).continuous_of_finiteDimensional |
| exact h.mul h.star |
|
|
| lemma positive_slice_image : {Q : M | 0 ≤ Q ∧ Matrix.trace Q = 1} = |
| flatGram '' Metric.sphere (0 : EuclideanSpace ℝ (ν × ν)) 1 := by |
| ext Q |
| constructor |
| · rintro ⟨hQ,htr⟩ |
| let x : EuclideanSpace ℝ (ν × ν) := WithLp.toLp 2 (fun p => CFC.sqrt Q p.1 p.2) |
| have hxQ : flatGram x = Q := by |
| change CFC.sqrt Q * star (CFC.sqrt Q) = Q |
| rw [(CFC.sqrt_nonneg Q).isSelfAdjoint.star_eq] |
| exact CFC.sqrt_mul_sqrt_self Q |
| have hn : ‖x‖ ^ 2 = 1 := by rw [← flatGram_trace,hxQ,htr] |
| have hnorm : ‖x‖ = 1 := by nlinarith [norm_nonneg x] |
| exact ⟨x,by simpa [dist_zero_right] using hnorm,hxQ⟩ |
| · rintro ⟨x,hx,rfl⟩ |
| refine ⟨flatGram_nonneg x,?_⟩ |
| have hnorm : ‖x‖ = 1 := by simpa [dist_zero_right] using hx |
| rw [flatGram_trace,hnorm,one_pow] |
|
|
| lemma positive_slice_compact : IsCompact {Q : M | 0 ≤ Q ∧ Matrix.trace Q = 1} := by |
| rw [positive_slice_image] |
| exact (isCompact_sphere (0 : EuclideanSpace ℝ (ν × ν)) 1).image flatGram_continuous |
|
|
| end PositiveSlice |
| end BorsukNine |
|
|
| end OAI |
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|
