| import OAI.Analysis.DirectCrouzeix.Duality |
| |
| namespace OAI |
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| universe u_46 |
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| noncomputable section |
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| open scoped Matrix Matrix.Norms.L2Operator Kronecker |
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| namespace DirectCrouzeix |
| |
| open scoped MatrixOrder ComplexOrder |
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| open MeasureTheory |
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| theorem norm_entry_le_l2_opNorm {ι : Type u_46} [Fintype ι] [DecidableEq ι] |
| (A : Matrix ι ι ℂ) (i j : ι) : ‖A i j‖ ≤ ‖A‖ := by |
| let e : EuclideanSpace ℂ ι := EuclideanSpace.single j 1 |
| let T := Matrix.toEuclideanCLM (n := ι) (𝕜 := ℂ) A |
| have he : ‖e‖ = 1 := by simp [e] |
| have hij : (T e) i = A i j := by |
| change (A *ᵥ Pi.single j 1) i = _ |
| rw [Matrix.mulVec_single_one] |
| rfl |
| calc |
| ‖A i j‖ = ‖(T e) i‖ := congrArg norm hij.symm |
| _ ≤ ‖T e‖ := PiLp.norm_apply_le _ _ |
| _ ≤ ‖T‖ * ‖e‖ := T.le_opNorm _ |
| _ = ‖A‖ := by simp [T, he, Matrix.l2_opNorm_toEuclideanCLM] |
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| def sharpnessMatrix : Matrix (Fin 2) (Fin 2) ℂ := !![0, 2 |
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| def sharpnessCoefficients : Fin 2 → Matrix (Fin 1) (Fin 1) ℂ := ![0, 1] |
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| theorem norm_le_one_of_mem_sharpnessRange {z : ℂ} |
| (hz : z ∈ numericalRange sharpnessMatrix) : ‖z‖ ≤ 1 := by |
| obtain ⟨u, hu, rfl⟩ := hz |
| have hsum : ‖u 0‖ ^ 2 + ‖u 1‖ ^ 2 = 1 := by |
| simpa [hu, Fin.sum_univ_two] using (EuclideanSpace.norm_sq_eq u).symm |
| have hev : inner ℂ u |
| (Matrix.toEuclideanCLM (n := Fin 2) (𝕜 := ℂ) sharpnessMatrix u) = |
| 2 * star (u 0) * u 1 := by |
| simp [sharpnessMatrix, EuclideanSpace.inner_eq_star_dotProduct, |
| Matrix.ofLp_toEuclideanCLM, dotProduct, Matrix.mulVec, Fin.sum_univ_two] |
| ring |
| rw [hev] |
| simp only [norm_mul, norm_star] |
| norm_num |
| nlinarith [sq_nonneg (‖u 0‖ - ‖u 1‖)] |
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| @[simp] theorem sharpness_polynomialValue (z : ℂ) : |
| polynomialValue (d := 1) sharpnessCoefficients z = z • 1 := by |
| simp [polynomialValue, sharpnessCoefficients, Fin.sum_univ_two] |
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| @[simp] theorem sharpness_tensorEvaluation : |
| tensorEvaluation (d := 1) sharpnessMatrix sharpnessCoefficients = |
| sharpnessMatrix ⊗ₖ (1 : Matrix (Fin 1) (Fin 1) ℂ) := by |
| simp [tensorEvaluation, sharpnessCoefficients, Fin.sum_univ_two] |
|
|
| theorem sharpness_rangeMaximum_le_one : |
| rangeMaximum (d := 1) sharpnessMatrix sharpnessCoefficients ≤ 1 := by |
| obtain ⟨z, hz, he⟩ := rangeMaximum_attained (by norm_num : 0 < 2) |
| sharpnessMatrix (d := 1) sharpnessCoefficients |
| rw [← he, sharpness_polynomialValue, norm_smul, norm_one, mul_one] |
| exact norm_le_one_of_mem_sharpnessRange hz |
|
|
| theorem sharpness_tensor_norm_ge_two : |
| 2 ≤ ‖tensorEvaluation (d := 1) sharpnessMatrix sharpnessCoefficients‖ := by |
| have h := norm_entry_le_l2_opNorm |
| (sharpnessMatrix ⊗ₖ (1 : Matrix (Fin 1) (Fin 1) ℂ)) (0, 0) (1, 0) |
| rw [sharpness_tensorEvaluation] |
| simpa [sharpnessMatrix, Matrix.kroneckerMap_apply] using h |
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|
| theorem uniform_sharpness (c : ℝ) (hc : UniversalBound c) : 2 ≤ c := by |
| have hmain := hc 2 1 1 (by norm_num) (by norm_num) sharpnessMatrix sharpnessCoefficients |
| have hmax := sharpness_rangeMaximum_le_one |
| have hlower := sharpness_tensor_norm_ge_two |
| have hnonneg : 0 ≤ rangeMaximum (d := 1) sharpnessMatrix sharpnessCoefficients := by |
| obtain ⟨z, hz, he⟩ := rangeMaximum_attained (by norm_num : 0 < 2) |
| sharpnessMatrix (d := 1) sharpnessCoefficients |
| rw [← he] |
| exact norm_nonneg _ |
| have hcpos : 0 ≤ c := by |
| by_contra hn |
| have := mul_nonpos_of_nonpos_of_nonneg (le_of_not_ge hn) hnonneg |
| linarith |
| have hmul := mul_le_mul_of_nonneg_left hmax hcpos |
| linarith |
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|
| end DirectCrouzeix |
| |
| end |
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| end OAI |
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