| import Mathlib | |
| import OAI.Probability.SKRatio.Matrices.EntryLM | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory InformationTheory Real Set | |
| open scoped NNReal ENNReal | |
| open Filter | |
| open scoped Topology | |
| noncomputable section | |
| open Matrix Real | |
| open scoped BigOperators Matrix.Norms.Frobenius ENNReal NNReal | |
| noncomputable section | |
| open Matrix Real | |
| open scoped BigOperators Matrix.Norms.Frobenius NNReal | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Real Set Filter | |
| open MeasureTheory.Measure | |
| open scoped ENNReal NNReal MeasureTheory Topology | |
| open MeasureTheory | |
| noncomputable section | |
| noncomputable section | |
| open MeasureTheory Set NormedSpace | |
| open scoped Topology | |
| noncomputable section | |
| open Matrix Real | |
| open scoped BigOperators Matrix.Norms.Frobenius | |
| namespace SKRatioGaussian.ComplexMatrix | |
| open MeasureTheory Set NormedSpace | |
| open scoped Topology FourierTransform SchwartzMap | |
| variable {ι : Type*} [Fintype ι] [DecidableEq ι] | |
| local instance : ContinuousENorm (Matrix ι ι ℂ) := | |
| @SeminormedAddGroup.toContinuousENorm _ Matrix.frobeniusSeminormedAddCommGroup.toSeminormedAddGroup | |
| local instance : TopologicalSpace.PseudoMetrizableSpace (Matrix ι ι ℂ) := | |
| inferInstanceAs (TopologicalSpace.PseudoMetrizableSpace (ι → ι → ℂ)) | |
| def linCLM : Matrix ι ι ℂ →L[ℂ] (EuclideanSpace ℂ ι →L[ℂ] EuclideanSpace ℂ ι) := | |
| (linEquiv (ι := ι)).toLinearMap.toContinuousLinearMap | |
| lemma opNorm_kernelMatrix {g : ℝ → ℂ} (hg : Integrable g) | |
| (M : Matrix ι ι ℂ) (hM : Mᴴ = M) : | |
| opNorm (kernelMatrix g M) ≤ ∫ t : ℝ, ‖g t‖ := by | |
| let : CompleteSpace (EuclideanSpace ℂ ι) := PiLp.completeSpace _ _ | |
| let : CompleteSpace (EuclideanSpace ℂ ι →L[ℂ] EuclideanSpace ℂ ι) := | |
| ContinuousLinearMap.instCompleteSpace | |
| have hi := linCLM.integral_comp_comm (kernel_integrable hg M hM) | |
| change ‖linCLM (∫ t : ℝ, g t • exp (imaginary t M))‖ ≤ _ | |
| calc | |
| _ = ‖∫ t : ℝ, linCLM (g t • exp (imaginary t M))‖ := congrArg norm hi.symm | |
| _ ≤ ∫ t : ℝ, ‖linCLM (g t • exp (imaginary t M))‖ := norm_integral_le_integral_norm _ | |
| _ ≤ ∫ t : ℝ, ‖g t‖ := by | |
| apply integral_mono (linCLM.integrable_comp (kernel_integrable hg M hM)).norm hg.norm | |
| intro t | |
| change ‖lin (g t • exp (imaginary t M))‖ ≤ _ | |
| rw [lin_csmul,norm_smul] | |
| exact (mul_le_mul_of_nonneg_left (imaginary_exp_opNorm_le t M hM) (norm_nonneg _)).trans_eq (mul_one _) | |
| lemma schwartzMatrix_opNorm (f : 𝓢(ℝ,ℂ)) (M : Matrix ι ι ℂ) (hM : Mᴴ = M) : | |
| opNorm (schwartzMatrix f M) ≤ ∫ t : ℝ, ‖(𝓕 f) t‖ := | |
| opNorm_kernelMatrix (𝓕 f).integrable _ (by simp [hM]) | |
| lemma schwartzMatrix_eq_inverse (f : 𝓢(ℝ,ℂ)) {lo hi : ℝ} (hlo : 0 < lo) | |
| (hf : ∀ x ∈ Icc lo hi, f x = (x : ℂ)⁻¹) | |
| (M : Matrix ι ι ℂ) (hM : M.IsHermitian) | |
| (hspec : ∀ i, hM.eigenvalues i ∈ Icc lo hi) : | |
| schwartzMatrix f M = M⁻¹ := by | |
| rw [schwartzMatrix_spectral f M hM] | |
| symm | |
| apply Matrix.inv_eq_left_inv | |
| conv_lhs => rhs; rw [hM.spectral_theorem] | |
| change conjugate hM.eigenvectorUnitary (diagonal (fun i => f (hM.eigenvalues i))) * | |
| conjugate hM.eigenvectorUnitary (diagonal (fun i => (hM.eigenvalues i : ℂ))) = 1 | |
| rw [← map_mul,diagonal_mul_diagonal] | |
| have hd : diagonal (fun i => f (hM.eigenvalues i) * (hM.eigenvalues i : ℂ)) = (1 : Matrix ι ι ℂ) := by | |
| have hz (i : ι) : (hM.eigenvalues i : ℂ) ≠ 0 := by exact_mod_cast ne_of_gt (lt_of_lt_of_le hlo (hspec i).1) | |
| simp_rw [hf _ (hspec _),inv_mul_cancel₀ (hz _)] | |
| exact diagonal_one | |
| rw [hd,map_one] | |
| lemma schwartzMatrix_hermitian (f : 𝓢(ℝ,ℂ)) (hf : ∀ x, star (f x) = f x) | |
| (M : Matrix ι ι ℂ) (hM : M.IsHermitian) : (schwartzMatrix f M).IsHermitian := by | |
| rw [schwartzMatrix_spectral f M hM] | |
| change star (conjugate hM.eigenvectorUnitary (diagonal _)) = _ | |
| rw [← map_star] | |
| congr 1 | |
| change (diagonal (fun i => f (hM.eigenvalues i)))ᴴ = _ | |
| rw [Matrix.diagonal_conjTranspose] | |
| congr 1 | |
| funext i | |
| exact hf (hM.eigenvalues i) | |
| end SKRatioGaussian.ComplexMatrix | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end OAI | |