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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 | import OAI.Analysis.HyperbolicCones.Model
namespace OAI
noncomputable section
open scoped BigOperators Matrix.Norms.L2Operator
open Matrix
universe u
namespace Paper256
theorem sym_isHermitian {n : ℕ} (X : Sym n) : (X : Mat n ℝ).IsHermitian := X.property
theorem qMatrix_smul {R : Type u} [CommRing R] (y : Fin 3 → R) (s : R) :
qMatrix (s • y) = s ^ 2 • qMatrix y := by
ext i j
by_cases h : i = j <;> simp [qMatrix, h, mul_pow] <;> ring
theorem phi_add {R : Type u} [CommRing R] (y : Fin 3 → R) (X Z : Mat 4 R) :
phi y (X + Z) = phi y X + phi y Z := by
ext i j
refine Fin.cases ?_ (fun i => ?_) i <;>
refine Fin.cases ?_ (fun j => ?_) j <;>
simp [phi, Matrix.trace, Matrix.mul_apply, Matrix.submatrix,
Finset.sum_add_distrib, mul_add, add_mul, add_comm]
theorem phi_smul {R : Type u} [CommRing R] (y : Fin 3 → R) (s : R) (X : Mat 4 R) :
phi y (s • X) = s • phi y X := by
ext i j
refine Fin.cases ?_ (fun i => ?_) i <;>
refine Fin.cases ?_ (fun j => ?_) j <;>
simp [phi, Matrix.trace, Matrix.mul_apply, Matrix.submatrix,
← Finset.mul_sum, mul_left_comm, mul_assoc]
theorem phi_zero {R : Type u} [CommRing R] (y : Fin 3 → R) : phi y (0 : Mat 4 R) = 0 := by
simpa using phi_smul y 0 (0 : Mat 4 R)
theorem phi_scaling_general {R : Type u} [CommRing R]
(y : Fin 3 → R) (s : R) (X : Mat 4 R) :
phi (s • y) X = s ^ 2 • phi y X := by
ext i j
refine Fin.cases ?_ (fun i => ?_) i <;>
refine Fin.cases ?_ (fun j => ?_) j <;>
simp [phi, qMatrix_smul, Matrix.trace, Matrix.mul_apply, Matrix.submatrix,
← Finset.mul_sum, mul_left_comm, mul_assoc]
theorem phi_scaling (y : Fin 3 → ℝ) (s : ℝ) (X : Mat 4 ℝ) :
phi (s • y) X = s ^ 2 • phi y X :=
phi_scaling_general y s X
theorem phi_zero_parameter {R : Type u} [CommRing R] (X : Mat 4 R) :
phi (0 : Fin 3 → R) X = 0 := by
simpa using phi_scaling_general (0 : Fin 3 → R) 0 X
theorem phi_neg_parameter {R : Type u} [CommRing R] (y : Fin 3 → R) (X : Mat 4 R) :
phi (-y) X = phi y X := by
simpa using phi_scaling_general y (-1) X
theorem phi_rank_one (y : Fin 3 → ℝ) (u v : Vec 4) :
dotProduct (fun i => u i) (phi y (outer v) *ᵥ (fun i => u i)) =
choiLam (wedgeCoordinates v u) y := by
simp [dotProduct, mulVec, phi, Matrix.trace, Matrix.mul_apply, Matrix.submatrix,
qMatrix, outer, wedgeCoordinates, choiLam, Fin.sum_univ_succ]
ring
theorem matrixValue_inverse {K : Type u} [Field K] (X Z : Mat 4 K) (y : Fin 3 → K)
(hX : IsUnit (Matrix.det X)) :
matrixValue X Z y = Matrix.det X ^ 4 * Matrix.det (Z - phi y X⁻¹) := by
have hadj : X.adjugate = X.det • X⁻¹ := by
rw [Matrix.inv_def, smul_smul, Ring.mul_inverse_cancel _ hX, one_smul]
rw [matrixValue, hadj, phi_smul, ← smul_sub, Matrix.det_smul]
rfl
end Paper256
end
end OAI
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