File size: 3,645 Bytes
c4cdbbd | 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 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 | import OAI.Probability.InvariantIsing.Cavity.CavityBaseEigenspaces
namespace OAI
/-! The base interaction has an explicit spectral decomposition. Its
retained multiplicities and the assigned special multiplicities add to
the required base counts. -/
noncomputable section
open scoped BigOperators Matrix
namespace InvariantIsing
lemma cavityBaseReplacement_spectral_sum {r s d : ℕ} {ι : Type*} [Fintype ι]
(J : Matrix (Fin r) (Fin r) ℝ) (W : Matrix (Fin r) (Fin s) ℝ)
(D : Matrix (Fin s) (Fin s) ℝ) (B : Matrix (Fin s) (Fin d) ℝ)
(A₀ : Matrix (Fin d) (Fin d) ℝ) (R : Matrix (Fin r) ι ℝ) (L : Matrix ι ι ℝ)
(hJR : J * R = R * L) (hJW : J * W = W * D)
(hcomplete : R * R.transpose + W * W.transpose = 1) :
cavityBaseReplacement J W D B A₀ =
R * L * R.transpose + (W * B) * A₀ * (W * B).transpose := by
have hJ : J = R * L * R.transpose + W * D * W.transpose := by
calc
J = J * (R * R.transpose + W * W.transpose) := by rw [hcomplete, Matrix.mul_one]
_ = _ := by rw [Matrix.mul_add, ← Matrix.mul_assoc, ← Matrix.mul_assoc, hJR, hJW]
unfold cavityBaseReplacement
rw [hJ]
abel
lemma cavityRetainedStack_perp {r m n : ℕ} (k : Fin m → ℕ)
(R : (a : Fin m) → Matrix (Fin r) (Fin (k a)) ℝ)
(X : Fin m → Matrix (Fin r) (Fin n) ℝ)
(hRX : ∀ a b, (R a).transpose * cavityNormalizeFrame (X b) = 0) :
(cavityRetainedStack k R).transpose * cavityEigenspaceFrame X = 0 := by
ext ⟨a, i⟩ j
exact congrArg (fun M : Matrix (Fin (k a)) (Fin n) ℝ =>
M i (finProdFinEquiv.symm j).2) (hRX a (finProdFinEquiv.symm j).1)
lemma cavityRetainedStack_eigen {r m : ℕ} (g : Fin r → Fin m) (lam : Fin m → ℝ)
(k : Fin m → ℕ) (R : (a : Fin m) → Matrix (Fin r) (Fin (k a)) ℝ)
(hR : ∀ a i, g i ≠ a → ∀ j, R a i j = 0) :
Matrix.diagonal (fun i => lam (g i)) * cavityRetainedStack k R =
cavityRetainedStack k R * Matrix.diagonal (fun j : Sigma (fun a : Fin m => Fin (k a)) => lam j.1) := by
ext i ⟨a, j⟩
have h := congrArg (fun M : Matrix (Fin r) (Fin (k a)) ℝ => M i j)
(cavitySpectralSupport_eigen g a lam (R a) (hR a))
rw [Matrix.mul_diagonal]
simpa only [Matrix.diagonal_mul, cavityRetainedStack, Matrix.smul_apply,
smul_eq_mul, mul_comm] using h
theorem cavityBase_spectral_decomposition {r m n d : ℕ}
(g : Fin r → Fin m) (A : Matrix (Fin r) (Fin n) ℝ) (lam : Fin m → ℝ)
(k : Fin m → ℕ) (R : (a : Fin m) → Matrix (Fin r) (Fin (k a)) ℝ)
(hR : ∀ a i, g i ≠ a → ∀ j, R a i j = 0)
(hproj : ∀ a, R a * (R a).transpose +
cavityNormalizeFrame (cavitySpectralImage g A a) *
(cavityNormalizeFrame (cavitySpectralImage g A a)).transpose =
Matrix.diagonal (fun i => if g i = a then (1 : ℝ) else 0))
(B : Matrix (Fin (m * n)) (Fin d) ℝ) (A₀ : Matrix (Fin d) (Fin d) ℝ) :
cavityBaseReplacement (Matrix.diagonal (fun i => lam (g i)))
(cavityEigenspaceFrame (cavitySpectralImage g A)) (cavityRepeatedSpectrum (n := n) lam) B A₀ =
cavityRetainedStack k R * Matrix.diagonal (fun j : Sigma (fun a : Fin m => Fin (k a)) => lam j.1) *
(cavityRetainedStack k R).transpose +
(cavityEigenspaceFrame (cavitySpectralImage g A) * B) * A₀ *
(cavityEigenspaceFrame (cavitySpectralImage g A) * B).transpose := by
apply cavityBaseReplacement_spectral_sum
· exact cavityRetainedStack_eigen g lam k R hR
· exact cavityEigenspaceFrame_eigen _ _ lam (cavitySpectralImage_eigen g A lam)
· exact cavityRetainedStack_complete g A k R hproj
end InvariantIsing
end
end OAI
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