| import OAI.Analysis.HyperbolicCones.MatrixBasic |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open scoped BigOperators Matrix.Norms.L2Operator MatrixOrder |
| open Matrix |
|
|
| universe u |
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| namespace Paper256 |
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|
| def phiLinear {R : Type u} [CommRing R] (y : Fin 3 → R) : Mat 4 R →ₗ[R] Mat 4 R where |
| toFun := phi y |
| map_add' := phi_add y |
| map_smul' := phi_smul y |
|
|
| theorem phi_isHermitian (y : Fin 3 → ℝ) (X : Mat 4 ℝ) (hX : X.IsHermitian) : |
| (phi y X).IsHermitian := by |
| have hQ (i j : Fin 3) : qMatrix y j i = qMatrix y i j := by |
| by_cases h : i = j |
| · subst j; rfl |
| · simp [qMatrix, h, Ne.symm h, mul_comm] |
| have hXs (i j : Fin 4) : X j i = X i j := by simpa using hX.apply i j |
| apply Matrix.IsHermitian.ext |
| intro i j |
| simp only [star_trivial] |
| refine Fin.cases ?_ (fun i => ?_) i <;> |
| refine Fin.cases ?_ (fun j => ?_) j |
| · rfl |
| · simp only [phi, Fin.cases_zero, Fin.cases_succ] |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro k _ |
| rw [hQ, hXs 0 k.succ] |
| ring |
| · simp only [phi, Fin.cases_zero, Fin.cases_succ] |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro k _ |
| rw [hQ, hXs 0 k.succ] |
| ring |
| · simp only [phi, Fin.cases_succ] |
| rw [hQ] |
|
|
| theorem posSemidef_sum_outer {n : ℕ} (X : Mat n ℝ) (hX : X.PosSemidef) : |
| ∃ v : Fin n → Vec n, X = ∑ i, outer (v i) := by |
| obtain ⟨B, hB⟩ := CStarAlgebra.nonneg_iff_eq_star_mul_self.mp hX.nonneg |
| refine ⟨fun i => WithLp.toLp 2 (B i), ?_⟩ |
| rw [hB] |
| ext i j |
| simp [Matrix.mul_apply, Matrix.sum_apply, outer] |
|
|
| end Paper256 |
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|
| end |
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| end OAI |
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|
