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ddc84c5 | 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 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 | import OAI.Analysis.DirectCrouzeix.Duality
namespace OAI
universe u_46
noncomputable section
open scoped Matrix Matrix.Norms.L2Operator Kronecker
namespace DirectCrouzeix
open scoped MatrixOrder ComplexOrder
open MeasureTheory
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]
def sharpnessMatrix : Matrix (Fin 2) (Fin 2) ℂ := !![0, 2; 0, 0]
def sharpnessCoefficients : Fin 2 → Matrix (Fin 1) (Fin 1) ℂ := ![0, 1]
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‖)]
@[simp] theorem sharpness_polynomialValue (z : ℂ) :
polynomialValue (d := 1) sharpnessCoefficients z = z • 1 := by
simp [polynomialValue, sharpnessCoefficients, Fin.sum_univ_two]
@[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
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
end DirectCrouzeix
end
end OAI
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