openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Spectral /PositiveRankOneQuadratic.lean
| import OAI.Probability.InvariantIsing.Spectral.PositiveRankOneInverse | |
| namespace OAI | |
| /-! The exact quadratic and trace increments of a positive rank-one resolvent update. -/ | |
| noncomputable section | |
| open Matrix | |
| namespace InvariantIsing | |
| lemma positive_rankOne_quadratic {N : ℕ} (A : Matrix (Fin N) (Fin N) ℝ) (hA : A.PosDef) | |
| (x : Fin N → ℝ) : | |
| x ⬝ᵥ ((A+vecMulVec x x)⁻¹ *ᵥ x) = (x ⬝ᵥ (A⁻¹ *ᵥ x))/(1+x ⬝ᵥ (A⁻¹ *ᵥ x)) := by | |
| let u := A⁻¹ *ᵥ x | |
| let q := x ⬝ᵥ u | |
| have hq : 0 < 1+q := by | |
| have hh := hA.inv.posSemidef.dotProduct_mulVec_nonneg x | |
| simp only [star_trivial] at hh | |
| change 0 ≤ q at hh | |
| linarith | |
| have hh : vecMulVec u u *ᵥ x = q • u := by | |
| ext i | |
| simp only [Matrix.mulVec,Matrix.vecMulVec_apply,dotProduct,Pi.smul_apply,smul_eq_mul] | |
| dsimp only [q] | |
| simp only [dotProduct] | |
| rw [Finset.sum_mul] | |
| apply Finset.sum_congr rfl | |
| intro j _ | |
| ring | |
| rw [positive_rankOne_inverse A hA x,Matrix.sub_mulVec,Matrix.smul_mulVec] | |
| change x ⬝ᵥ (u-(1+q)⁻¹ • (vecMulVec u u *ᵥ x)) = q/(1+q) | |
| rw [hh,dotProduct_sub,dotProduct_smul,dotProduct_smul] | |
| change q-(1+q)⁻¹*(q*q) = q/(1+q) | |
| field_simp [hq.ne'] | |
| ring | |
| lemma positive_rankOne_trace_difference {N : ℕ} (A : Matrix (Fin N) (Fin N) ℝ) (hA : A.PosDef) | |
| (x : Fin N → ℝ) : | |
| A⁻¹.trace-(A+vecMulVec x x)⁻¹.trace = | |
| ((A⁻¹ *ᵥ x) ⬝ᵥ (A⁻¹ *ᵥ x))/(1+x ⬝ᵥ (A⁻¹ *ᵥ x)) := by | |
| rw [positive_rankOne_inverse A hA x,Matrix.trace_sub,Matrix.trace_smul,Matrix.trace_vecMulVec] | |
| simp only [smul_eq_mul] | |
| ring | |
| end InvariantIsing | |
| end | |
| end OAI | |