| import Mathlib |
| import OAI.Probability.BinarySweep.MatrixBounds.TraceInterpolation |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators Matrix.Norms.L2Operator |
|
|
| namespace BinaryCoordinateSweeps.TraceHolder |
| variable {ι : Type*} [Fintype ι] [DecidableEq ι] |
|
|
| def matrixMoment (q : ℕ) (A : M ι) : ℝ := (Matrix.trace ((star A * A)^q)).re |
|
|
| lemma diagReal_pow (s : ι → ℝ) (q : ℕ) : |
| (diagReal s)^q = diagReal (fun i => s i ^ q) := by |
| simp [diagReal, Matrix.diagonal_pow] |
|
|
| lemma trace_unitary_conj (V : Matrix.unitaryGroup ι ℂ) (B : M ι) : |
| Matrix.trace ((V : M ι) * B * star (V : M ι)) = Matrix.trace B := by |
| rw [Matrix.trace_mul_cycle, Unitary.coe_star_mul_self, one_mul] |
|
|
| lemma unitary_conj_pow (V : Matrix.unitaryGroup ι ℂ) (B : M ι) (q : ℕ) : |
| ((V : M ι) * B * star (V : M ι))^q = (V : M ι) * B^q * star (V : M ι) := by |
| have h := ((Unitary.conjStarAlgAut ℂ (M ι) V).toAlgEquiv.map_pow B q).symm |
| change (Unitary.conjStarAlgAut ℂ (M ι) V B)^q = |
| Unitary.conjStarAlgAut ℂ (M ι) V (B^q) at h |
| simpa only [Unitary.conjStarAlgAut_apply, Unitary.coe_star] using h |
|
|
| lemma matrixMoment_of_svd (A : M ι) (q : ℕ) (V : Matrix.unitaryGroup ι ℂ) |
| (s : ι → ℝ) (hgram : star A * A = (V : M ι) * diagReal (fun i => s i^2) * star (V : M ι)) : |
| matrixMoment q A = ∑ i, s i ^ (2*q) := by |
| rw [matrixMoment, hgram, unitary_conj_pow, trace_unitary_conj, diagReal_pow] |
| simp only [diagReal, Matrix.trace_diagonal, |
| Complex.re_sum, Complex.ofReal_re, ← pow_mul] |
|
|
| lemma matrixMoment_nonneg (q : ℕ) (A : M ι) : 0 ≤ matrixMoment q A := by |
| obtain ⟨U, V, s, hs, hU, hA, hg⟩ := exists_partial_svd A |
| rw [matrixMoment_of_svd A q V s hg] |
| exact Finset.sum_nonneg (fun i _ => pow_nonneg (hs i) _) |
|
|
| lemma matrixMoment_eq_zero_iff {q : ℕ} (hq : 0 < q) (A : M ι) : |
| matrixMoment q A = 0 ↔ A = 0 := by |
| constructor |
| · intro h |
| obtain ⟨U, V, s, hs, hU, hA, hg⟩ := exists_partial_svd A |
| rw [matrixMoment_of_svd A q V s hg] at h |
| have hi : ∀ i, s i = 0 := by |
| intro i |
| have hz := (Finset.sum_eq_zero_iff_of_nonneg (fun j _ => pow_nonneg (hs j) (2*q))).mp h |
| exact (pow_eq_zero_iff (by omega : 2*q ≠ 0)).mp (hz i (Finset.mem_univ _)) |
| rw [hA] |
| simp [diagReal, hi] |
| · intro h |
| simp [matrixMoment, h, zero_pow (by omega : q ≠ 0)] |
|
|
| lemma unitary_norm_le_one (V : Matrix.unitaryGroup ι ℂ) : ‖(V : M ι)‖ ≤ 1 := by |
| have h1 : ‖(1 : M ι)‖ ≤ 1 := by |
| rw [← Matrix.diagonal_one, Matrix.l2_opNorm_diagonal] |
| exact (pi_norm_le_iff_of_nonneg zero_le_one).mpr (by simp) |
| have h : ‖(V : M ι)‖ * ‖(V : M ι)‖ ≤ 1 := by |
| rw [← CStarRing.norm_star_mul_self, Unitary.coe_star_mul_self] |
| exact h1 |
| nlinarith [norm_nonneg (V : M ι)] |
|
|
| omit [Fintype ι] in |
| lemma diagPower_real (d : ι → ℝ) (hd : ∀ i, 0 ≤ d i) {x : ℝ} (hx : 0 < x) : |
| diagPower d (x : ℂ) = diagReal (fun i => (d i) ^ x) := by |
| unfold diagPower diagReal |
| congr 1 |
| funext i |
| by_cases hi : d i = 0 |
| · simp [hi, Real.zero_rpow hx.ne'] |
| · rw [ite_eq_right hi] |
| change Complex.exp ((x : ℂ) * (Real.log (d i) : ℂ)) = ((d i ^ x : ℝ) : ℂ) |
| rw [Real.rpow_def_of_pos (lt_of_le_of_ne (hd i) (Ne.symm hi))] |
| rw [← Complex.ofReal_mul, ← Complex.ofReal_exp] |
| congr 2 |
| ring |
|
|
| lemma density_normalization {m : ℕ} (hm : 0 < m) (s : ι → ℝ) (hs : ∀ i, 0 ≤ s i) |
| {t : ℝ} (ht : 0 < t) (hts : ∑ i, s i^m = t) : |
| let d : ι → ℝ := fun i => s i ^ m / t |
| (∀ i, 0 ≤ d i) ∧ (∑ i, d i = 1) ∧ |
| diagReal s = (t ^ ((m : ℝ)⁻¹) : ℝ) • diagPower d (((m : ℝ)⁻¹ : ℝ) : ℂ) := by |
| dsimp only |
| have hm' : 0 < (m : ℝ) := by exact_mod_cast hm |
| have hd : ∀ i, 0 ≤ s i^m/t := fun i => div_nonneg (pow_nonneg (hs i) _) ht.le |
| refine ⟨hd, ?_, ?_⟩ |
| · rw [← Finset.sum_div, hts, div_self ht.ne'] |
| · rw [diagPower_real _ hd (inv_pos.mpr hm')] |
| unfold diagReal |
| rw [← Matrix.diagonal_smul] |
| congr 1 |
| funext i |
| change (s i : ℂ) = ((t ^ (m : ℝ)⁻¹ : ℝ) : ℂ) * |
| (((s i ^ m / t) ^ (m : ℝ)⁻¹ : ℝ) : ℂ) |
| rw [Real.div_rpow (pow_nonneg (hs i) _) ht.le, |
| Real.pow_rpow_inv_natCast (hs i) (Nat.ne_of_gt hm)] |
| norm_cast |
| exact (mul_div_cancel₀ (s i) (Real.rpow_pos_of_pos ht _).ne').symm |
|
|
| end BinaryCoordinateSweeps.TraceHolder |
|
|
| end |
|
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| end OAI |
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|
