openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Harmonic /GeneralizedGramSpectrum.lean
| import Mathlib | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open scoped Classical BigOperators Matrix InnerProductSpace | |
| namespace CriticalPercolation.HarmonicAnalysis | |
| variable {ι : Type*} [Fintype ι] | |
| def matrixQuadratic (M : Matrix ι ι ℝ) (v : ι → ℝ) : ℝ := v ⬝ᵥ M *ᵥ v | |
| lemma hermitian_dot_comm (M : Matrix ι ι ℝ) (hM : M.IsHermitian) (x y : ι → ℝ) : | |
| x ⬝ᵥ M *ᵥ y = y ⬝ᵥ M *ᵥ x := by | |
| have ht : Mᵀ=M := Matrix.isHermitian_iff_isSymm.mp hM | |
| calc | |
| _ = Matrix.vecMul x M ⬝ᵥ y := Matrix.dotProduct_mulVec x M y | |
| _ = Matrix.vecMul x Mᵀ ⬝ᵥ y := by rw [ht] | |
| _ = _ := by rw [Matrix.vecMul_transpose,dotProduct_comm] | |
| theorem generalized_gram_spectrum (A B : Matrix ι ι ℝ) (hA : A.PosDef) | |
| (hB : B.IsHermitian) (hmono : ∀ x, matrixQuadratic A x≤ matrixQuadratic B x) : | |
| ∃ (μ : Fin (Fintype.card ι) → ℝ) (b : Module.Basis (Fin (Fintype.card ι)) ℝ (ι → ℝ)), | |
| (∀ i, 1≤μ i) ∧ B.det=A.det*(∏ i, μ i) ∧ | |
| (∀ x, matrixQuadratic A x=∑ i, (b.repr x i)^2) ∧ | |
| (∀ x, matrixQuadratic B x=∑ i, μ i*(b.repr x i)^2) := by | |
| let := A.toNormedAddCommGroup hA | |
| let : SeminormedAddCommGroup (ι → ℝ) := (A.toNormedAddCommGroup hA).toSeminormedAddCommGroup | |
| let := A.toInnerProductSpace hA.posSemidef | |
| let T : (ι → ℝ) →ₗ[ℝ] (ι → ℝ) := Matrix.toLin' (A⁻¹*B) | |
| have hdet : IsUnit A.det := isUnit_iff_ne_zero.mpr hA.det_pos.ne' | |
| have hinner (x y : ι → ℝ) : ⟪x,y⟫_ℝ=x ⬝ᵥ A *ᵥ y := by | |
| change (A *ᵥ y) ⬝ᵥ star x = _ | |
| simp only [star_trivial,dotProduct_comm] | |
| have hTinner (x y : ι → ℝ) : ⟪x,T y⟫_ℝ=x ⬝ᵥ B *ᵥ y := by | |
| rw [hinner] | |
| change x ⬝ᵥ A *ᵥ ((A⁻¹*B) *ᵥ y)=_ | |
| rw [Matrix.mulVec_mulVec,Matrix.mul_nonsing_inv_cancel_left A B hdet] | |
| have hT : LinearMap.IsSymmetric (𝕜 := ℝ) (E := ι → ℝ) T := by | |
| intro x y | |
| rw [real_inner_comm y (T x),hTinner,hTinner] | |
| exact hermitian_dot_comm B hB y x | |
| have hn : Module.finrank ℝ (ι → ℝ)=Fintype.card ι := Module.finrank_fintype_fun_eq_card ℝ | |
| let b := hT.eigenvectorBasis hn | |
| let μ := hT.eigenvalues hn | |
| have hQa (x : ι → ℝ) : matrixQuadratic A x=⟪x,x⟫_ℝ := (hinner x x).symm | |
| have hQb (x : ι → ℝ) : matrixQuadratic B x=⟪x,T x⟫_ℝ := (hTinner x x).symm | |
| have hμ (i : Fin (Fintype.card ι)) : 1≤μ i := by | |
| have hh := hmono (b i) | |
| rw [hQa,hQb,hT.apply_eigenvectorBasis hn i,inner_smul_right] at hh | |
| change ⟪b i,b i⟫_ℝ ≤ μ i*⟪b i,b i⟫_ℝ at hh | |
| simpa only [real_inner_self_eq_norm_sq,OrthonormalBasis.norm_eq_one,one_pow,mul_one] using hh | |
| refine ⟨μ,b.toBasis,hμ,?_,?_,?_⟩ | |
| · have hh := hT.det_eq_prod_eigenvalues hn | |
| change (Matrix.toLin' (A⁻¹*B)).det=∏ i, μ i at hh | |
| rw [LinearMap.det_toLin',Matrix.det_mul,Matrix.det_nonsing_inv,Ring.inverse_eq_inv] at hh | |
| calc | |
| B.det = A.det*(A.det⁻¹*B.det) := by field_simp [hA.det_pos.ne'] | |
| _ = _ := congrArg (A.det*·) hh | |
| · intro x | |
| rw [hQa,←b.sum_inner_mul_inner x x] | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| rw [b.coe_toBasis_repr_apply,b.repr_apply_apply,real_inner_comm x (b i)] | |
| ring | |
| · intro x | |
| rw [hQb,←b.sum_inner_mul_inner x (T x)] | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| rw [b.coe_toBasis_repr_apply,b.repr_apply_apply,←hT (b i) x, | |
| hT.apply_eigenvectorBasis hn i,inner_smul_left,real_inner_comm x (b i)] | |
| change ⟪x,b i⟫_ℝ * (μ i*⟪x,b i⟫_ℝ) = μ i*⟪x,b i⟫_ℝ^2 | |
| ring | |
| end CriticalPercolation.HarmonicAnalysis | |
| end | |
| end | |
| end OAI | |