File size: 2,381 Bytes
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 | import OAI.Probability.InvariantIsing.Spectral.PositiveResolventSpectrum
import Mathlib.Analysis.InnerProductSpace.Trace
namespace OAI
/-! The positive resolvent in the eigenbasis of the original positive matrix. -/
noncomputable section
open Matrix
open scoped BigOperators RealInnerProductSpace
namespace InvariantIsing
lemma positiveResolvent_eigenvector {N : ℕ} {t : ℝ} (ht : 0 < t)
{B : Matrix (Fin N) (Fin N) ℝ} (hB : B.PosSemidef) (i : Fin N) :
(positiveResolvent t B).toEuclideanLin (hB.isHermitian.eigenvectorBasis i) =
(t+hB.isHermitian.eigenvalues i)⁻¹ • hB.isHermitian.eigenvectorBasis i := by
let u := (hB.isHermitian.eigenvectorBasis i).ofLp
let a := hB.isHermitian.eigenvalues i
have ha : 0 < t+a := add_pos_of_pos_of_nonneg ht (hB.eigenvalues_nonneg i)
have hBu : B *ᵥ u = a • u := hB.isHermitian.mulVec_eigenvectorBasis i
have hAu : (t • (1 : Matrix (Fin N) (Fin N) ℝ)+B) *ᵥ u = (t+a) • u := by
rw [Matrix.add_mulVec,Matrix.smul_mulVec,Matrix.one_mulVec,hBu,add_smul]
have hinv : positiveResolvent t B*(t • (1 : Matrix (Fin N) (Fin N) ℝ)+B) = 1 :=
Matrix.nonsing_inv_mul _ (isUnit_iff_ne_zero.mpr
(((Matrix.PosDef.one.smul ht).add_posSemidef hB).det_pos.ne'))
have hh : (t+a) • (positiveResolvent t B *ᵥ u) = u := by
rw [← Matrix.mulVec_smul,← hAu,Matrix.mulVec_mulVec,hinv,Matrix.one_mulVec]
have hu : positiveResolvent t B *ᵥ u = (t+a)⁻¹ • u := by
calc
_ = (t+a)⁻¹ • ((t+a) • (positiveResolvent t B *ᵥ u)) := by
rw [smul_smul,inv_mul_cancel₀ ha.ne',one_smul]
_ = _ := by rw [hh]
ext j
exact congrFun hu j
lemma positiveResolvent_trace_eigenvalues {N : ℕ} {t : ℝ} (ht : 0 < t)
{B : Matrix (Fin N) (Fin N) ℝ} (hB : B.PosSemidef) :
(positiveResolvent t B).trace = ∑ i, (t+hB.isHermitian.eigenvalues i)⁻¹ := by
have htr := LinearMap.trace_eq_sum_inner (positiveResolvent t B).toEuclideanLin hB.isHermitian.eigenvectorBasis
rw [Matrix.toEuclideanLin_eq_toLin_orthonormal,Matrix.trace_toLin_eq] at htr
rw [htr]
apply Finset.sum_congr rfl
intro i _
rw [← Matrix.toEuclideanLin_eq_toLin_orthonormal]
rw [positiveResolvent_eigenvector ht hB,real_inner_smul_right,real_inner_self_eq_norm_sq]
simp only [hB.isHermitian.eigenvectorBasis.orthonormal.1 i,one_pow,mul_one]
end InvariantIsing
end
end OAI
|