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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 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 99 100 101 | import OAI.Analysis.HyperbolicCones.MatrixComplex
import OAI.Analysis.HyperbolicCones.FormPositive
namespace OAI
noncomputable section
open scoped BigOperators Matrix.Norms.L2Operator MatrixOrder ComplexOrder
open Matrix
universe u
namespace Paper256
theorem qMatrix_transpose {R : Type u} [CommRing R] (y : Fin 3 → R) :
(qMatrix y)ᵀ = qMatrix y := by
ext i j
by_cases h : i = j
· subst j; rfl
· simp [qMatrix, h, Ne.symm h, mul_comm]
theorem phi_transpose {R : Type u} [CommRing R] (y : Fin 3 → R) (X : Mat 4 R) :
phi y Xᵀ = (phi y X)ᵀ := by
have hQ (i j : Fin 3) : qMatrix y j i = qMatrix y i j :=
congrFun (congrFun (qMatrix_transpose y) i) j
ext i j
refine Fin.cases ?_ (fun i => ?_) i <;>
refine Fin.cases ?_ (fun j => ?_) j
· change Matrix.trace (qMatrix y * (X.submatrix Fin.succ Fin.succ)ᵀ) =
Matrix.trace (qMatrix y * X.submatrix Fin.succ Fin.succ)
conv_lhs => rw [← qMatrix_transpose y]
rw [← Matrix.transpose_mul, Matrix.trace_transpose, Matrix.trace_mul_comm]
· simp only [phi, Fin.cases_zero, Fin.cases_succ, Matrix.transpose_apply]
congr 1
apply Finset.sum_congr rfl
intro k _
rw [hQ]
ring
· simp only [phi, Fin.cases_zero, Fin.cases_succ, Matrix.transpose_apply]
congr 1
apply Finset.sum_congr rfl
intro k _
rw [hQ]
ring
· simp only [phi, Fin.cases_succ, Matrix.transpose_apply]
rw [hQ]
theorem phi_conjTranspose (y : Fin 3 → ℝ) (X : Mat 4 ℂ) :
(phi (fun i => (y i : ℂ)) X)ᴴ = phi (fun i => (y i : ℂ)) Xᴴ := by
change ((phi (fun i => (y i : ℂ)) X)ᵀ).map (starRingEnd ℂ) = _
rw [← phi_transpose, phi_map]
have hy : (fun i => (starRingEnd ℂ) (y i : ℂ)) = (fun i => (y i : ℂ)) :=
funext fun i => Complex.conj_ofReal (y i)
rw [hy]
rfl
theorem phi_complex_rank_one (y : Fin 3 → ℝ) (u v : Fin 4 → ℂ) :
dotProduct (star u)
(phi (fun i => (y i : ℂ)) (vecMulVec v (star v)) *ᵥ u) =
dotProduct (star (fun i : Fin 3 => v 0 * u i.succ - u 0 * v i.succ))
(qMatrix (fun i => (y i : ℂ)) *ᵥ
(fun i : Fin 3 => v 0 * u i.succ - u 0 * v i.succ)) := by
simp [dotProduct, mulVec, phi, qMatrix, Matrix.trace, Matrix.mul_apply,
Matrix.submatrix, vecMulVec, Fin.sum_univ_succ]
ring
theorem phi_complex_outer_posSemidef (y : Fin 3 → ℝ) (v : Fin 4 → ℂ) :
(phi (fun i => (y i : ℂ)) (vecMulVec v (star v))).PosSemidef := by
have hq : (qMatrix (fun i => (y i : ℂ))).PosSemidef := by
have h := posSemidef_complexify (qMatrix y) (qMatrix_posSemidef y)
change ((qMatrix y).map (algebraMap ℝ ℂ)).PosSemidef at h
rw [qMatrix_map] at h
exact h
apply Matrix.PosSemidef.of_dotProduct_mulVec_nonneg
· show (phi (fun i => (y i : ℂ)) (vecMulVec v (star v)))ᴴ = _
rw [phi_conjTranspose, (posSemidef_vecMulVec_self_star v).isHermitian.eq]
· intro u
rw [phi_complex_rank_one]
exact hq.dotProduct_mulVec_nonneg _
theorem phi_complex_posSemidef (y : Fin 3 → ℝ) (X : Mat 4 ℂ) (hX : X.PosSemidef) :
(phi (fun i => (y i : ℂ)) X).PosSemidef := by
obtain ⟨B, hB⟩ := CStarAlgebra.nonneg_iff_eq_star_mul_self.mp hX.nonneg
have hx : X = ∑ i, vecMulVec (star (B i)) (B i) := by
rw [hB]
ext i j
simp [Matrix.mul_apply, Matrix.sum_apply, vecMulVec]
rw [hx]
change ((phiLinear (fun i => (y i : ℂ))) (∑ i, vecMulVec (star (B i)) (B i))).PosSemidef
rw [map_sum]
apply Matrix.posSemidef_sum
intro i _
change (phi (fun i => (y i : ℂ)) (vecMulVec (star (B i)) (B i))).PosSemidef
simpa only [star_star] using phi_complex_outer_posSemidef y (star (B i))
end Paper256
end
end OAI
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