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a458723 | 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 | import Mathlib
import OAI.Analysis.Conductivity.Model
namespace OAI
noncomputable section
namespace ScalarConductivity
open Set Matrix Filter Topology
open scoped Matrix.Norms.Elementwise
variable {ι P : Type*} [Fintype ι] [DecidableEq ι] [TopologicalSpace P]
omit [DecidableEq ι] in
lemma matrix_pi_mulVec_bound (A : Matrix ι ι ℝ) (b : ι → ℝ) (i : ι) :
‖(A*ᵥb) i‖≤(Fintype.card ι:ℝ)*‖A‖*‖b‖ := by
change ‖∑ j,A i j*b j‖≤_
calc
_≤∑ j,‖A i j*b j‖ := norm_sum_le _ _
_≤∑ j,‖A‖*‖b‖ := by
apply Finset.sum_le_sum
intro j _
rw [norm_mul]
exact mul_le_mul ((norm_le_pi_norm (A i) j).trans (norm_le_pi_norm A i))
(norm_le_pi_norm b j) (norm_nonneg _) (norm_nonneg _)
_=_ := by simp [mul_assoc]
theorem matrix_inverse_locally_bounded {M : P → Matrix ι ι ℝ} {p₀ : P}
(hM : ContinuousAt M p₀) (h₀ : IsUnit (M p₀)) :
∃ C : ℝ,0<C ∧ ∀ᶠ p in 𝓝 p₀,
IsUnit (M p) ∧ (∀ b : ι → ℝ,
M p*ᵥ((M p)⁻¹*ᵥb)=b ∧ ∀ i,‖((M p)⁻¹*ᵥb) i‖≤C*‖b‖) := by
have hd₀ : (M p₀).det≠0 := isUnit_iff_ne_zero.mp ((isUnit_iff_isUnit_det _).mp h₀)
have hdi : ContinuousAt Ring.inverse (M p₀).det := by
simpa only [Ring.inverse_eq_inv'] using (continuousAt_inv₀ hd₀)
have hi : ContinuousAt (fun p => (M p)⁻¹) p₀ :=
(continuousAt_matrix_inv _ hdi).comp hM
have hn : ∀ᶠ p in 𝓝 p₀,‖(M p)⁻¹‖<‖(M p₀)⁻¹‖+1 :=
hi.norm.eventually (gt_mem_nhds (by linarith))
have hd : ∀ᶠ p in 𝓝 p₀,(M p).det≠0 :=
((continuous_id.matrix_det).continuousAt.comp hM).tendsto.eventually (isOpen_ne.mem_nhds hd₀)
let C := ((Fintype.card ι:ℝ)+1)*(‖(M p₀)⁻¹‖+1)
have hC : 0<C := by dsimp [C]; positivity
refine ⟨C,hC,?_⟩
filter_upwards [hn,hd] with p hp hpd
have hu : IsUnit (M p).det := isUnit_iff_ne_zero.mpr hpd
refine ⟨(isUnit_iff_isUnit_det _).mpr hu,?_⟩
intro b
refine ⟨?_,?_⟩
· rw [Matrix.mulVec_mulVec,Matrix.mul_nonsing_inv _ hu,Matrix.one_mulVec]
· intro i
apply (matrix_pi_mulVec_bound _ b i).trans
apply mul_le_mul_of_nonneg_right _ (norm_nonneg b)
have hn₀ : 0≤‖(M p₀)⁻¹‖+1 := by positivity
dsimp only [C]
have hh := mul_le_mul_of_nonneg_left hp.le (show (0:ℝ)≤Fintype.card ι by positivity)
nlinarith
end ScalarConductivity
end
end OAI
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