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a89d282 | 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 | import OAI.Probability.InvariantIsing.Haar.HaarKernelGeometry
namespace OAI
/-! Unnormalized Frobenius geometry for the averaging and tangent estimates. -/
noncomputable section
open Matrix
open scoped BigOperators Matrix.Norms.Frobenius
namespace InvariantIsing
lemma frobenius_norm_sq {N : ℕ} (A : Matrix (Fin N) (Fin N) ℝ) :
‖A‖^2 = matrixFrobeniusPair A A := by
rw [Matrix.frobenius_norm_def]
simp only [Real.rpow_two,Real.norm_eq_abs,sq_abs]
have hs : 0 ≤ ∑ i : Fin N, ∑ j : Fin N, (A i j)^2 :=
Finset.sum_nonneg fun i _ => Finset.sum_nonneg fun j _ => sq_nonneg _
rw [← Real.sqrt_eq_rpow,Real.sq_sqrt hs,matrixFrobeniusPair_self]
lemma frobeniusDistance_eq_norm {N : ℕ} (U V : SpecialOrthogonal N) :
frobeniusDistance U V = ‖(U : Matrix (Fin N) (Fin N) ℝ)-V‖ := by
rw [Matrix.frobenius_norm_def]
simp only [Real.rpow_two,Real.norm_eq_abs,sq_abs,Matrix.sub_apply]
exact Real.sqrt_eq_rpow _
lemma frobenius_norm_mul_orthogonal {N : ℕ} (A : Matrix (Fin N) (Fin N) ℝ)
(U : SpecialOrthogonal N) : ‖A*(U : Matrix (Fin N) (Fin N) ℝ)‖ = ‖A‖ := by
apply (sq_eq_sq₀ (norm_nonneg _) (norm_nonneg _)).mp
rw [frobenius_norm_sq,frobenius_norm_sq]
unfold matrixFrobeniusPair
rw [Matrix.transpose_mul,Matrix.trace_mul_cycle]
rw [mul_assoc A U (U : Matrix (Fin N) (Fin N) ℝ).transpose,
(Matrix.mem_orthogonalGroup_iff (Fin N) ℝ).mp
(Matrix.mem_specialOrthogonalGroup_iff.mp U.property).1,mul_one,Matrix.trace_mul_comm]
lemma frobeniusDistance_mul_left {N : ℕ} (W U V : SpecialOrthogonal N) :
frobeniusDistance (W*U) (W*V) = frobeniusDistance U V := by
have h := congrArg (fun x : ℝ => 2*N-x) (haarKernelBase_mul_left W U V)
rw [haarKernelBase_dist,haarKernelBase_dist] at h
apply (sq_eq_sq₀ (Real.sqrt_nonneg _) (Real.sqrt_nonneg _)).mp
change (frobeniusDistance (W*U) (W*V))^2 = (frobeniusDistance U V)^2
linarith
lemma frobeniusDistance_comm {N : ℕ} (U V : SpecialOrthogonal N) :
frobeniusDistance U V = frobeniusDistance V U := by
rw [frobeniusDistance_eq_norm,frobeniusDistance_eq_norm,← norm_neg,neg_sub]
lemma frobeniusDistance_mul_right {N : ℕ} (U V W : SpecialOrthogonal N) :
frobeniusDistance (U*W) (V*W) = frobeniusDistance U V := by
rw [frobeniusDistance_eq_norm,frobeniusDistance_eq_norm]
change ‖(U : Matrix (Fin N) (Fin N) ℝ)*W-(V : Matrix (Fin N) (Fin N) ℝ)*W‖ = _
rw [← sub_mul,frobenius_norm_mul_orthogonal]
end InvariantIsing
end
end OAI
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