| import Mathlib | |
| namespace OAI | |
| noncomputable section | |
| namespace Problem340 | |
| theorem eigenpair_modulus_le_of_norm_bound | |
| {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V] | |
| (A : V →ₗ[ℝ] V) (C : ℝ) (hC : 0 ≤ C) | |
| (hA : ∀ x, ‖A x‖ ≤ C * ‖x‖) | |
| (a b : ℝ) (u v : V) (hne : u ≠ 0 ∨ v ≠ 0) | |
| (hu : A u = a • u - b • v) | |
| (hv : A v = b • u + a • v) : | |
| Real.sqrt (a ^ 2 + b ^ 2) ≤ 2 * C := by | |
| have hab : 0 ≤ a ^ 2 + b ^ 2 := add_nonneg (sq_nonneg a) (sq_nonneg b) | |
| have hu' : (a ^ 2 + b ^ 2) • u = a • A u + b • A v := by | |
| rw [hu, hv] | |
| module | |
| have hv' : (a ^ 2 + b ^ 2) • v = (-b) • A u + a • A v := by | |
| rw [hu, hv] | |
| module | |
| have hnu : (a ^ 2 + b ^ 2) * ‖u‖ ≤ |a| * ‖A u‖ + |b| * ‖A v‖ := by | |
| calc | |
| (a ^ 2 + b ^ 2) * ‖u‖ = ‖(a ^ 2 + b ^ 2) • u‖ := by | |
| rw [norm_smul, Real.norm_eq_abs, abs_of_nonneg hab] | |
| _ = ‖a • A u + b • A v‖ := congrArg norm hu' | |
| _ ≤ ‖a • A u‖ + ‖b • A v‖ := norm_add_le _ _ | |
| _ = |a| * ‖A u‖ + |b| * ‖A v‖ := by simp [norm_smul] | |
| have hnv : (a ^ 2 + b ^ 2) * ‖v‖ ≤ |b| * ‖A u‖ + |a| * ‖A v‖ := by | |
| calc | |
| (a ^ 2 + b ^ 2) * ‖v‖ = ‖(a ^ 2 + b ^ 2) • v‖ := by | |
| rw [norm_smul, Real.norm_eq_abs, abs_of_nonneg hab] | |
| _ = ‖(-b) • A u + a • A v‖ := congrArg norm hv' | |
| _ ≤ ‖(-b) • A u‖ + ‖a • A v‖ := norm_add_le _ _ | |
| _ = |b| * ‖A u‖ + |a| * ‖A v‖ := by simp [norm_smul] | |
| have hS : 0 < ‖u‖ + ‖v‖ := by | |
| rcases hne with hu0 | hv0 | |
| · exact add_pos_of_pos_of_nonneg (norm_pos_iff.mpr hu0) (norm_nonneg v) | |
| · exact add_pos_of_nonneg_of_pos (norm_nonneg u) (norm_pos_iff.mpr hv0) | |
| have hsum : (a ^ 2 + b ^ 2) * (‖u‖ + ‖v‖) ≤ | |
| ((|a| + |b|) * C) * (‖u‖ + ‖v‖) := by | |
| have h1 := mul_le_mul_of_nonneg_left (hA u) (abs_nonneg a) | |
| have h2 := mul_le_mul_of_nonneg_left (hA v) (abs_nonneg b) | |
| have h3 := mul_le_mul_of_nonneg_left (hA u) (abs_nonneg b) | |
| have h4 := mul_le_mul_of_nonneg_left (hA v) (abs_nonneg a) | |
| nlinarith | |
| have hbound : a ^ 2 + b ^ 2 ≤ (|a| + |b|) * C := | |
| le_of_mul_le_mul_right hsum hS | |
| have ha : |a| ≤ Real.sqrt (a ^ 2 + b ^ 2) := by | |
| exact Real.abs_le_sqrt (by nlinarith [sq_nonneg b]) | |
| have hb : |b| ≤ Real.sqrt (a ^ 2 + b ^ 2) := by | |
| exact Real.abs_le_sqrt (by nlinarith [sq_nonneg a]) | |
| have hsqrt := Real.sq_sqrt hab | |
| have hroot := Real.sqrt_nonneg (a ^ 2 + b ^ 2) | |
| nlinarith [mul_le_mul_of_nonneg_right (add_le_add ha hb) hC] | |
| theorem bddAbove_eigenpair_moduli_of_norm_bound | |
| {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V] | |
| (A : V →ₗ[ℝ] V) (C : ℝ) (hC : 0 ≤ C) | |
| (hA : ∀ x, ‖A x‖ ≤ C * ‖x‖) : | |
| BddAbove {r : ℝ | ∃ a b : ℝ, ∃ u v : V, | |
| (u ≠ 0 ∨ v ≠ 0) ∧ A u = a • u - b • v ∧ | |
| A v = b • u + a • v ∧ r = Real.sqrt (a ^ 2 + b ^ 2)} := by | |
| refine ⟨2 * C, ?_⟩ | |
| rintro r ⟨a, b, u, v, hne, hu, hv, rfl⟩ | |
| exact eigenpair_modulus_le_of_norm_bound A C hC hA a b u v hne hu hv | |
| theorem bddAbove_eigenpair_moduli_continuousLinearMap | |
| {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V] | |
| (A : V →L[ℝ] V) : | |
| BddAbove {r : ℝ | ∃ a b : ℝ, ∃ u v : V, | |
| (u ≠ 0 ∨ v ≠ 0) ∧ A u = a • u - b • v ∧ | |
| A v = b • u + a • v ∧ r = Real.sqrt (a ^ 2 + b ^ 2)} := by | |
| exact bddAbove_eigenpair_moduli_of_norm_bound A.toLinearMap ‖A‖ | |
| (norm_nonneg A) A.le_opNorm | |
| end Problem340 | |
| end | |
| end OAI | |