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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 | import OAI.Probability.InvariantIsing.Gaussian.HermitianGaussianQuadratic
import OAI.Probability.InvariantIsing.Gaussian.GaussianGramResolvent
namespace OAI
/-! Spectral bounds for the positive resolvents occurring after deletion of a column. -/
noncomputable section
open Matrix
open scoped BigOperators RealInnerProductSpace
namespace InvariantIsing
def positiveResolvent {N : ℕ} (t : ℝ) (B : Matrix (Fin N) (Fin N) ℝ) : Matrix (Fin N) (Fin N) ℝ :=
(t • (1 : Matrix (Fin N) (Fin N) ℝ)+B)⁻¹
lemma positiveResolvent_posDef {N : ℕ} {t : ℝ} (ht : 0 < t)
{B : Matrix (Fin N) (Fin N) ℝ} (hB : B.PosSemidef) : (positiveResolvent t B).PosDef :=
((Matrix.PosDef.one.smul ht).add_posSemidef hB).inv
lemma positiveResolvent_eigenvalues {N : ℕ} {t : ℝ} (ht : 0 < t)
{B : Matrix (Fin N) (Fin N) ℝ} (hB : B.PosSemidef) (i : Fin N) :
0 ≤ (positiveResolvent_posDef ht hB).isHermitian.eigenvalues i ∧
(positiveResolvent_posDef ht hB).isHermitian.eigenvalues i ≤ 1/t := by
let R := positiveResolvent t B
let hR : R.PosDef := positiveResolvent_posDef ht hB
let u := hR.isHermitian.eigenvectorBasis i
let a := hR.isHermitian.eigenvalues i
have hu : u.ofLp ⬝ᵥ u.ofLp = 1 := by
change ⟪u,u⟫ = 1
rw [real_inner_self_eq_norm_sq]
simp only [u,hR.isHermitian.eigenvectorBasis.orthonormal.1 i,one_pow]
have ha : 0 ≤ a := by
dsimp only [a]
rw [hR.isHermitian.eigenvalues_eq]
simpa only [star_trivial,RCLike.re_to_real] using hR.posSemidef.dotProduct_mulVec_nonneg u.ofLp
have hq : 0 ≤ u.ofLp ⬝ᵥ (B *ᵥ u.ofLp) := by
simpa only [star_trivial] using hB.dotProduct_mulVec_nonneg u.ofLp
have hinv : (t • (1 : Matrix (Fin N) (Fin N) ℝ)+B)*R = 1 :=
Matrix.mul_nonsing_inv _ (isUnit_iff_ne_zero.mpr
(((Matrix.PosDef.one.smul ht).add_posSemidef hB).det_pos.ne'))
have heig : R *ᵥ u.ofLp = a • u.ofLp := hR.isHermitian.mulVec_eigenvectorBasis i
have hh : (t • (1 : Matrix (Fin N) (Fin N) ℝ)+B) *ᵥ (R *ᵥ u.ofLp) = u.ofLp := by
rw [Matrix.mulVec_mulVec,hinv,Matrix.one_mulVec]
rw [heig,Matrix.mulVec_smul,Matrix.add_mulVec,Matrix.smul_mulVec,Matrix.one_mulVec] at hh
have hd := congrArg (fun y => u.ofLp ⬝ᵥ y) hh
rw [dotProduct_smul,dotProduct_add,dotProduct_smul,hu] at hd
simp only [smul_eq_mul,mul_one] at hd
constructor
· exact ha
· apply (le_div_iff₀ ht).mpr
nlinarith [mul_nonneg ha hq]
end InvariantIsing
end
end OAI
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