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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 | 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
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