File size: 3,278 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 | import OAI.Probability.InvariantIsing.Cavity.CavityBaseReplacement
import OAI.Probability.InvariantIsing.Cavity.CavityRetainedStack
namespace OAI
/-! The replaced interaction preserves retained eigenvectors, gives the
special axes their assigned eigenvalues, and annihilates the cavity. -/
noncomputable section
open scoped Matrix
namespace InvariantIsing
lemma cavityBaseReplacement_retained {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) (hWR : W.transpose * R = 0) :
cavityBaseReplacement J W D B A₀ * R = R * L := by
simp only [cavityBaseReplacement, Matrix.add_mul, Matrix.sub_mul, Matrix.transpose_mul,
Matrix.mul_assoc, hWR, Matrix.mul_zero, sub_zero, add_zero, hJR]
lemma cavityBaseReplacement_special {r s d : ℕ}
(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)
(hJW : J * W = W * D) (hW : W.transpose * W = 1) (hB : B.transpose * B = 1) :
cavityBaseReplacement J W D B A₀ * (W * B) = (W * B) * A₀ := by
have hJB : J * (W * B) = W * D * B := by rw [← Matrix.mul_assoc, hJW]
have hWB : W.transpose * (W * B) = B := by rw [← Matrix.mul_assoc, hW, Matrix.one_mul]
have hBB : (W * B).transpose * (W * B) = 1 := by
rw [Matrix.transpose_mul, Matrix.mul_assoc, hWB, hB]
rw [cavityBaseReplacement, Matrix.add_mul, Matrix.sub_mul, hJB,
Matrix.mul_assoc (W * D) W.transpose (W * B), hWB, sub_self, zero_add,
Matrix.mul_assoc (W * B * A₀) (W * B).transpose (W * B), hBB, Matrix.mul_one]
lemma cavityBaseReplacement_cavity {r s d n : ℕ}
(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)
(E : Matrix (Fin s) (Fin n) ℝ) (A₀ : Matrix (Fin d) (Fin d) ℝ)
(hJW : J * W = W * D) (hW : W.transpose * W = 1) (hBE : B.transpose * E = 0) :
cavityBaseReplacement J W D B A₀ * (W * E) = 0 := by
have hJE : J * (W * E) = W * D * E := by rw [← Matrix.mul_assoc, hJW]
have hWE : W.transpose * (W * E) = E := by rw [← Matrix.mul_assoc, hW, Matrix.one_mul]
have hBB : (W * B).transpose * (W * E) = 0 := by
rw [Matrix.transpose_mul, Matrix.mul_assoc, hWE, hBE]
rw [cavityBaseReplacement, Matrix.add_mul, Matrix.sub_mul, hJE,
Matrix.mul_assoc (W * D) W.transpose (W * E), hWE, sub_self, zero_add,
Matrix.mul_assoc (W * B * A₀) (W * B).transpose (W * E), hBB, Matrix.mul_zero]
lemma cavityBaseReplacement_conjugate {r s d : ℕ}
(J U : 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) :
cavityBaseReplacement (U * J * U.transpose) (U * W) D B A₀ =
U * cavityBaseReplacement J W D B A₀ * U.transpose := by
simp only [cavityBaseReplacement, Matrix.transpose_mul, Matrix.mul_add,
Matrix.add_mul, Matrix.mul_sub, Matrix.sub_mul, Matrix.mul_assoc]
end InvariantIsing
end
end OAI
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