openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Spectral /PositiveResolventEigenbasis.lean
| 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 | |