File size: 1,187 Bytes
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 | import OAI.Analysis.HyperbolicCones.MatrixLinear
import OAI.Analysis.HyperbolicCones.FormPositive
namespace OAI
noncomputable section
open scoped BigOperators Matrix.Norms.L2Operator MatrixOrder
open Matrix
namespace Paper256
theorem phi_outer_posSemidef (y : Fin 3 → ℝ) (v : Vec 4) :
(phi y (outer v)).PosSemidef := by
apply Matrix.PosSemidef.of_dotProduct_mulVec_nonneg
(phi_isHermitian y (outer v) (outerSym v).property)
intro u
have hr := phi_rank_one y (WithLp.toLp 2 u) v
simpa only [star_trivial, hr] using choiLam_nonnegative
(wedgeCoordinates v (WithLp.toLp 2 u)) y
theorem phi_posSemidef (y : Fin 3 → ℝ) (X : Mat 4 ℝ) (hX : X.PosSemidef) :
(phi y X).PosSemidef := by
obtain ⟨v, rfl⟩ := posSemidef_sum_outer X hX
change ((phiLinear y) (∑ i, outer (v i))).PosSemidef
rw [map_sum]
exact Matrix.posSemidef_sum _ fun i _ => phi_outer_posSemidef y (v i)
theorem phi_monotone (y : Fin 3 → ℝ) : Monotone (phi y) := by
intro X Z hXZ
have hp := phi_posSemidef y (Z - X) hXZ
change (phi y Z - phi y X).PosSemidef
change ((phiLinear y) (Z - X)).PosSemidef at hp
rw [map_sub] at hp
exact hp
end Paper256
end
end OAI
|