| import OAI.Analysis.DirectCrouzeix.Resolvent | |
| namespace OAI | |
| noncomputable section | |
| open scoped Matrix Matrix.Norms.L2Operator Kronecker | |
| namespace DirectCrouzeix | |
| open scoped MatrixOrder ComplexOrder | |
| def exteriorCorrection (h q : ℂ → ℂ) (ζ μ : ℂ) : ℂ := | |
| q ζ / (h ζ - h μ) - ζ / (ζ - μ) | |
| def exteriorKernel (h q : ℂ → ℂ) (ζ μ : ℂ) : ℝ := | |
| 1 + (exteriorCorrection h q ζ μ).re + (star (exteriorCorrection h q ζ μ)).re | |
| theorem re_circle_fraction {ζ μ : ℂ} (hζ : ‖ζ‖ = 1) (hμ : ‖μ‖ = 1) | |
| (hne : ζ ≠ μ) : 2 * (ζ / (ζ - μ)).re = 1 := by | |
| have hnζ : ζ.re ^ 2 + ζ.im ^ 2 = 1 := by | |
| simpa [Complex.normSq_apply, pow_two] using | |
| (show Complex.normSq ζ = 1 by rw [Complex.normSq_eq_norm_sq, hζ]; norm_num) | |
| have hnμ : μ.re ^ 2 + μ.im ^ 2 = 1 := by | |
| simpa [Complex.normSq_apply, pow_two] using | |
| (show Complex.normSq μ = 1 by rw [Complex.normSq_eq_norm_sq, hμ]; norm_num) | |
| have hd : Complex.normSq (ζ - μ) ≠ 0 := by | |
| exact mt Complex.normSq_eq_zero.mp (sub_ne_zero.mpr hne) | |
| rw [Complex.div_re] | |
| field_simp | |
| simp [Complex.normSq_apply, Complex.sub_re, Complex.sub_im] at * | |
| nlinarith | |
| theorem exteriorKernel_nonneg_off_diagonal (h q : ℂ → ℂ) | |
| {ζ μ : ℂ} (hζ : ‖ζ‖ = 1) (hμ : ‖μ‖ = 1) (hne : ζ ≠ μ) | |
| (hs : 0 ≤ (star (q ζ) * (h ζ - h μ)).re) : | |
| 0 ≤ exteriorKernel h q ζ μ := by | |
| have hfrac := re_circle_fraction hζ hμ hne | |
| have hp : 0 ≤ (q ζ / (h ζ - h μ)).re := by | |
| rw [Complex.div_re, ← add_div] | |
| apply div_nonneg _ (Complex.normSq_nonneg _) | |
| simpa [Complex.mul_re] using hs | |
| simp only [exteriorKernel, exteriorCorrection, Complex.sub_re, Complex.star_def, | |
| Complex.conj_re] | |
| linarith | |
| def coefficientEmbedding (n m : ℕ) : | |
| Matrix (Fin m) (Fin m) ℂ →+* | |
| Matrix (Fin n × Fin m) (Fin n × Fin m) ℂ where | |
| toFun B := (1 : Matrix (Fin n) (Fin n) ℂ) ⊗ₖ B | |
| map_zero' := Matrix.kronecker_zero _ | |
| map_one' := Matrix.one_kronecker_one | |
| map_add' B C := Matrix.kronecker_add _ B C | |
| map_mul' B C := by | |
| simpa using Matrix.mul_kronecker_mul | |
| (1 : Matrix (Fin n) (Fin n) ℂ) 1 B C | |
| def tensorPolynomial {n m : ℕ} (A : Matrix (Fin n) (Fin n) ℂ) : | |
| Polynomial (Matrix (Fin m) (Fin m) ℂ) →+* | |
| Matrix (Fin n × Fin m) (Fin n × Fin m) ℂ := | |
| Polynomial.eval₂RingHom' (coefficientEmbedding n m) | |
| (A ⊗ₖ (1 : Matrix (Fin m) (Fin m) ℂ)) (by | |
| intro B | |
| change (1 ⊗ₖ B) * (A ⊗ₖ 1) = (A ⊗ₖ 1) * (1 ⊗ₖ B) | |
| rw [← Matrix.mul_kronecker_mul, ← Matrix.mul_kronecker_mul] | |
| simp) | |
| def coefficientPolynomial {m d : ℕ} | |
| (B : Fin (d + 1) → Matrix (Fin m) (Fin m) ℂ) : | |
| Polynomial (Matrix (Fin m) (Fin m) ℂ) := | |
| ∑ k : Fin (d + 1), Polynomial.C (B k) * Polynomial.X ^ (k : ℕ) | |
| theorem kronecker_pow_one {n m : ℕ} (A : Matrix (Fin n) (Fin n) ℂ) (k : ℕ) : | |
| (A ⊗ₖ (1 : Matrix (Fin m) (Fin m) ℂ)) ^ k = (A ^ k) ⊗ₖ 1 := by | |
| induction k with | |
| | zero => simp | |
| | succ k ih => | |
| rw [pow_succ, ih, ← Matrix.mul_kronecker_mul] | |
| simp [pow_succ] | |
| @[simp] theorem tensorPolynomial_C {n m : ℕ} (A : Matrix (Fin n) (Fin n) ℂ) | |
| (B : Matrix (Fin m) (Fin m) ℂ) : | |
| tensorPolynomial A (Polynomial.C B) = (1 : Matrix (Fin n) (Fin n) ℂ) ⊗ₖ B := by | |
| exact Polynomial.eval₂_C _ _ | |
| @[simp] theorem tensorPolynomial_X {n m : ℕ} (A : Matrix (Fin n) (Fin n) ℂ) : | |
| tensorPolynomial (m := m) A Polynomial.X = A ⊗ₖ (1 : Matrix (Fin m) (Fin m) ℂ) := by | |
| exact Polynomial.eval₂_X _ _ | |
| theorem tensorPolynomial_coefficientPolynomial {n m d : ℕ} | |
| (A : Matrix (Fin n) (Fin n) ℂ) | |
| (B : Fin (d + 1) → Matrix (Fin m) (Fin m) ℂ) : | |
| tensorPolynomial A (coefficientPolynomial B) = tensorEvaluation A B := by | |
| simp only [coefficientPolynomial, map_sum, map_mul, map_pow, | |
| tensorPolynomial_C, tensorPolynomial_X, kronecker_pow_one, | |
| ← Matrix.mul_kronecker_mul, Matrix.one_mul, Matrix.mul_one, tensorEvaluation] | |
| theorem tensorPolynomial_mul {n m : ℕ} (A : Matrix (Fin n) (Fin n) ℂ) | |
| (F G : Polynomial (Matrix (Fin m) (Fin m) ℂ)) : | |
| tensorPolynomial A (F * G) = tensorPolynomial A F * tensorPolynomial A G := | |
| map_mul _ _ _ | |
| end DirectCrouzeix | |
| end | |
| end OAI | |