| 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 | |