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f205175 | 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 72 73 74 75 76 77 78 79 80 81 | 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
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