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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 | import OAI.Analysis.DirectCrouzeix.Model
namespace OAI
noncomputable section
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
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