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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 | import OAI.Analysis.HyperbolicCones.PencilBlocks
import OAI.Analysis.HyperbolicCones.RescalingBasic
namespace OAI
noncomputable section
open Set Matrix
open scoped Matrix.Norms.L2Operator
namespace Paper256.BlockPencil
theorem G_eq_zero {K : Set Ambient} (P : BlockPencil K)
(hSlice : ∀ (X Z : Sym 4) (y : Fin 3 → ℝ), (X : Mat 4 ℝ).PosDef →
(((X, Z), y) ∈ K ↔ ((Z : Mat 4 ℝ) - phi y (X : Mat 4 ℝ)⁻¹).PosSemidef))
(y : Fin 3 → ℝ) : P.G y = 0 := by
let Z0 : Sym 4 := ⟨phi y (1 : Mat 4 ℝ), phi_isHermitian y 1 Matrix.isHermitian_one⟩
apply linear_term_zero_of_quadratic_posSemidef (P.E Z0) (P.G y)
intro s
have hm : (((1, s ^ 2 • Z0), s • y) : Ambient) ∈ K := by
apply (hSlice 1 (s ^ 2 • Z0) (s • y) Matrix.PosDef.one).mpr
simp [Z0, phi_scaling]
exact Matrix.PosSemidef.zero
have hp := (P.represents 1 (s ^ 2 • Z0) (s • y)).mp hm
have hb := hp.submatrix Sum.inr
convert hb using 1
ext i j
simp [Matrix.submatrix, Matrix.fromBlocks]
theorem scaled_isHermitian {K : Set Ambient} (P : BlockPencil K)
(X Z : Sym 4) (y : Fin 3 → ℝ) (s : ℝ) : (P.scaled X Z y s).IsHermitian := by
apply Matrix.IsHermitian.fromBlocks
· exact ((P.D X).property.add ((IsSelfAdjoint.all s).smul (P.A y).property)).add
((IsSelfAdjoint.all (s ^ 2)).smul (P.F Z).property)
· simp
· exact (P.E Z).property
theorem scaled_zero {K : Set Ambient} (P : BlockPencil K)
(X Z : Sym 4) (y : Fin 3 → ℝ) : P.scaled X Z y 0 = P.limit X Z y := by
simp [scaled, limit]
end Paper256.BlockPencil
end
end OAI
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