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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 | import OAI.Probability.InvariantIsing.Spectral.PositiveResolventSpectrum
namespace OAI
/-! A dimension-free quadratic contraction for positive resolvents. -/
noncomputable section
open Matrix
open scoped BigOperators
namespace InvariantIsing
lemma positiveResolvent_quadratic_contraction {N : ℕ} {t : ℝ} (ht : 0 < t)
{B : Matrix (Fin N) (Fin N) ℝ} (hB : B.PosSemidef) (x : Fin N → ℝ) :
t*((positiveResolvent t B *ᵥ x) ⬝ᵥ (positiveResolvent t B *ᵥ x)) ≤
x ⬝ᵥ (positiveResolvent t B *ᵥ x) := by
let y := positiveResolvent t B *ᵥ x
have hinv : (t • (1 : Matrix (Fin N) (Fin N) ℝ)+B)*positiveResolvent t B = 1 :=
Matrix.mul_nonsing_inv _ (isUnit_iff_ne_zero.mpr
(((Matrix.PosDef.one.smul ht).add_posSemidef hB).det_pos.ne'))
have hy : (t • (1 : Matrix (Fin N) (Fin N) ℝ)+B) *ᵥ y = x := by
change (t • (1 : Matrix (Fin N) (Fin N) ℝ)+B) *ᵥ (positiveResolvent t B *ᵥ x) = x
rw [Matrix.mulVec_mulVec,hinv,Matrix.one_mulVec]
have hp : 0 ≤ y ⬝ᵥ (B *ᵥ y) := by
simpa only [star_trivial] using hB.dotProduct_mulVec_nonneg y
have hh := congrArg (fun z => y ⬝ᵥ z) hy
rw [Matrix.add_mulVec,Matrix.smul_mulVec,Matrix.one_mulVec,
dotProduct_add,dotProduct_smul] at hh
simp only [smul_eq_mul] at hh
rw [dotProduct_comm x y]
change t*(y ⬝ᵥ y) ≤ y ⬝ᵥ x
linarith
end InvariantIsing
end
end OAI
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