| import OAI.Analysis.DirectCrouzeix.Model |
|
|
| namespace OAI |
|
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| noncomputable section |
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| open scoped Matrix Matrix.Norms.L2Operator Kronecker |
|
|
| namespace DirectCrouzeix |
|
|
| theorem numericalRange_eq_image {n : ℕ} (A : Matrix (Fin n) (Fin n) ℂ) : |
| numericalRange A = |
| (fun u : EuclideanSpace ℂ (Fin n) => inner ℂ u |
| (Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℂ) A u)) '' Metric.sphere 0 1 := by |
| ext z |
| simp [numericalRange, Set.mem_image] |
| |
| theorem isCompact_numericalRange {n : ℕ} (A : Matrix (Fin n) (Fin n) ℂ) : |
| IsCompact (numericalRange A) := by |
| rw [numericalRange_eq_image] |
| exact (isCompact_sphere (0 : EuclideanSpace ℂ (Fin n)) 1).image |
| (continuous_id.inner (Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℂ) A).continuous) |
| |
| theorem numericalRange_nonempty {n : ℕ} (hn : 0 < n) |
| (A : Matrix (Fin n) (Fin n) ℂ) : (numericalRange A).Nonempty := by |
| let : NeZero n := ⟨Nat.ne_of_gt hn⟩ |
| rw [numericalRange_eq_image] |
| exact (NormedSpace.sphere_nonempty.mpr (by norm_num : (0 : ℝ) ≤ 1)).image _ |
| |
| theorem norm_le_of_mem_numericalRange {n : ℕ} (A : Matrix (Fin n) (Fin n) ℂ) |
| {z : ℂ} (hz : z ∈ numericalRange A) : ‖z‖ ≤ ‖A‖ := by |
| obtain ⟨u, hu, rfl⟩ := hz |
| calc |
| ‖inner ℂ u (Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℂ) A u)‖ |
| ≤ ‖u‖ * ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℂ) A u‖ := |
| norm_inner_le_norm _ _ |
| _ ≤ ‖u‖ * (‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℂ) A‖ * ‖u‖) := |
| mul_le_mul_of_nonneg_left (ContinuousLinearMap.le_opNorm _ _) (norm_nonneg _) |
| _ = ‖A‖ := by rw [hu, Matrix.l2_opNorm_toEuclideanCLM]; ring |
| |
| theorem continuous_polynomialValue {m d : ℕ} |
| (B : Fin (d + 1) → Matrix (Fin m) (Fin m) ℂ) : Continuous (polynomialValue B) := by |
| unfold polynomialValue |
| exact continuous_finsetSum _ (fun k _ => (continuous_id.pow _).smul continuous_const) |
| |
| theorem rangeMaximum_attained {n m d : ℕ} (hn : 0 < n) |
| (A : Matrix (Fin n) (Fin n) ℂ) |
| (B : Fin (d + 1) → Matrix (Fin m) (Fin m) ℂ) : |
| ∃ z ∈ numericalRange A, ‖polynomialValue B z‖ = rangeMaximum A B := by |
| have hc := (isCompact_numericalRange A).image (continuous_polynomialValue B).norm |
| have hne := (numericalRange_nonempty hn A).image (fun z => ‖polynomialValue B z‖) |
| obtain ⟨z, hz, he⟩ := hc.sSup_mem hne |
| exact ⟨z, hz, he⟩ |
| |
| theorem norm_polynomialValue_le_rangeMaximum {n m d : ℕ} |
| (A : Matrix (Fin n) (Fin n) ℂ) |
| (B : Fin (d + 1) → Matrix (Fin m) (Fin m) ℂ) |
| {z : ℂ} (hz : z ∈ numericalRange A) : ‖polynomialValue B z‖ ≤ rangeMaximum A B := by |
| apply le_csSup |
| · exact (isCompact_numericalRange A).bddAbove_image |
| (continuous_polynomialValue B).norm.continuousOn |
| · exact ⟨z, hz, rfl⟩ |
| |
| open scoped MatrixOrder ComplexOrder |
| |
| end DirectCrouzeix |
| |
| end |
| |
| end OAI |
| |
