| import OAI.Analysis.HyperbolicCones.PencilBlocks |
| import OAI.Analysis.HyperbolicCones.RescalingBasic |
|
|
| namespace OAI |
|
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| noncomputable section |
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|
| 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 |
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| end |
|
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| end OAI |
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|
