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9eeb9c7 | 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 | import OAI.Analysis.HyperbolicCones.MatrixBasic
namespace OAI
noncomputable section
open scoped BigOperators Matrix.Norms.L2Operator MatrixOrder
open Matrix
universe u
namespace Paper256
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
end
end OAI
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