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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 | import OAI.Probability.InvariantIsing.Gaussian.GaussianGramResolvent
namespace OAI
/-! The positive rank-one inverse identity used by the Gaussian leave-one-column resolvent. -/
noncomputable section
open Matrix
namespace InvariantIsing
lemma positive_rankOne_inverse {N : ℕ} (A : Matrix (Fin N) (Fin N) ℝ) (hA : A.PosDef)
(x : Fin N → ℝ) :
(A+vecMulVec x x)⁻¹ = A⁻¹-(1+x ⬝ᵥ (A⁻¹ *ᵥ x))⁻¹ •
vecMulVec (A⁻¹ *ᵥ x) (A⁻¹ *ᵥ x) := by
let R := A⁻¹
let u := R *ᵥ x
let q := x ⬝ᵥ u
have hq : 0 < 1+q := by
have h := hA.inv.posSemidef.dotProduct_mulVec_nonneg x
simp only [star_trivial] at h
change 0 ≤ q at h
linarith
have hR : Rᵀ = R := by
simpa only [R,Matrix.IsHermitian,Matrix.conjTranspose_eq_transpose_of_trivial] using hA.inv.isHermitian
have hxR : x ᵥ* R = u := by
calc
_ = x ᵥ* Rᵀ := congrArg (fun M => x ᵥ* M) hR.symm
_ = _ := Matrix.vecMul_transpose R x
have hAR : A*R = 1 := Matrix.mul_nonsing_inv A (isUnit_iff_ne_zero.mpr hA.det_pos.ne')
have hAx : A *ᵥ u = x := by
change A *ᵥ (R *ᵥ x) = x
rw [Matrix.mulVec_mulVec,hAR,Matrix.one_mulVec]
have h₁ : (A+vecMulVec x x)*R = 1+vecMulVec x u := by
rw [add_mul,hAR,Matrix.vecMulVec_mul,hxR]
have h₂ : (A+vecMulVec x x)*vecMulVec u u = (1+q) • vecMulVec x u := by
rw [add_mul,Matrix.mul_vecMulVec,hAx,Matrix.vecMulVec_mul_vecMulVec,Matrix.vecMulVec_smul]
ext i j
simp only [Matrix.add_apply,Matrix.smul_apply,Matrix.vecMulVec_apply,smul_eq_mul]
dsimp only [q]
ring
apply Matrix.inv_eq_right_inv
change (A+vecMulVec x x)*(R-(1+q)⁻¹ • vecMulVec u u) = 1
rw [mul_sub,mul_smul_comm,h₁,h₂,smul_smul,inv_mul_cancel₀ hq.ne',one_smul]
exact add_sub_cancel_right 1 (vecMulVec x u)
end InvariantIsing
end
end OAI
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