| 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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|
