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caec230 | 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 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 | import Mathlib
import OAI.MathematicalPhysics.PEPSFilters.LocalOperators
namespace OAI
/-! Common eigenbases for positive commuting matrices. -/
noncomputable section
open scoped BigOperators ComplexOrder
open scoped BigOperators ComplexOrder Matrix.Norms.L2Operator
open scoped BigOperators
open scoped Topology
open Filter
open scoped MatrixOrder
open PolynomialPEPS.PinnedEntropy
open scoped BigOperators Matrix.Norms.L2Operator
namespace PolynomialPEPS.Subvolume.JointDiagonalization
variable {ι : Type*} [Fintype ι] [DecidableEq ι]
theorem exists_subordinate {κ E : Type*} [NormedAddCommGroup E]
[InnerProductSpace ℂ E] [FiniteDimensional ℂ E]
(V : κ → Submodule ℂ E)
(horth : OrthogonalFamily ℂ (fun i => V i) (fun i => (V i).subtypeₗᵢ))
(htop : (⨆ i, V i) = ⊤) :
∃ (b : OrthonormalBasis (Fin (Module.finrank ℂ E)) ℂ E)
(idx : Fin (Module.finrank ℂ E) → κ), ∀ i, b i ∈ V (idx i) := by
classical
let : Fintype {i : κ // V i ≠ ⊥} := horth.independent.fintypeNeBotOfFiniteDimensional
let W : {i : κ // V i ≠ ⊥} → Submodule ℂ E := fun i => V i.val
have horthW := horth.comp (f := fun i : {i : κ // V i ≠ ⊥} => i.val) Subtype.val_injective
have htopW : (⨆ i, W i) = ⊤ := (iSup_ne_bot_subtype V).trans htop
have hV : DirectSum.IsInternal W := horthW.isInternal_iff.mpr
(by rw [htopW,Submodule.top_orthogonal_eq_bot])
refine ⟨hV.subordinateOrthonormalBasis rfl horthW,
(fun i => (hV.subordinateOrthonormalBasisIndex rfl i horthW).val),?_⟩
exact fun i => hV.subordinateOrthonormalBasis_subordinate rfl i horthW
theorem exists_basis (A B : Matrix ι ι ℂ) (hA : A.IsHermitian)
(hB : B.IsHermitian) (hAB : Commute A B) :
∃ (b : OrthonormalBasis ι ℂ (EuclideanSpace ℂ ι)) (d e : ι → ℂ),
(∀ i, A.toEuclideanLin (b i) = d i • b i) ∧
(∀ i, B.toEuclideanLin (b i) = e i • b i) := by
classical
have hA' := Matrix.isSymmetric_toEuclideanLin_iff.mpr hA
have hB' := Matrix.isSymmetric_toEuclideanLin_iff.mpr hB
have hAB' : Commute A.toEuclideanLin B.toEuclideanLin := by
change (Matrix.toLpLin 2 2 A).comp (Matrix.toLpLin 2 2 B) =
(Matrix.toLpLin 2 2 B).comp (Matrix.toLpLin 2 2 A)
rw [← Matrix.toLpLin_mul_same,← Matrix.toLpLin_mul_same,hAB.eq]
let V : (ℂ × ℂ) → Submodule ℂ (EuclideanSpace ℂ ι) :=
fun i => Module.End.eigenspace A.toEuclideanLin i.2 ⊓
Module.End.eigenspace B.toEuclideanLin i.1
have horth := hA'.orthogonalFamily_eigenspace_inf_eigenspace hB'
have htop : (⨆ i, V i) = ⊤ := by
rw [iSup_prod]
change (⨆ i : ℂ, ⨆ j : ℂ, Module.End.eigenspace A.toEuclideanLin j ⊓
Module.End.eigenspace B.toEuclideanLin i) = ⊤
rw [iSup_comm]
exact hA'.iSup_iSup_eigenspace_inf_eigenspace_eq_top_of_commute hB' hAB'
obtain ⟨b₀,idx,hm⟩ := exists_subordinate V horth htop
let eι : Fin (Module.finrank ℂ (EuclideanSpace ℂ ι)) ≃ ι :=
Fintype.equivOfCardEq (by simp)
refine ⟨b₀.reindex eι,(fun i => (idx (eι.symm i)).2),
(fun i => (idx (eι.symm i)).1),?_,?_⟩
· intro i
have h := Module.End.mem_eigenspace_iff.mp (hm (eι.symm i)).1
simpa only [OrthonormalBasis.reindex_apply] using h
· intro i
have h := Module.End.mem_eigenspace_iff.mp (hm (eι.symm i)).2
simpa only [OrthonormalBasis.reindex_apply] using h
def basisUnitary (b : OrthonormalBasis ι ℂ (EuclideanSpace ℂ ι)) :
unitary (Matrix ι ι ℂ) :=
⟨(EuclideanSpace.basisFun ι ℂ).toBasis.toMatrix b.toBasis,
(EuclideanSpace.basisFun ι ℂ).toMatrix_orthonormalBasis_mem_unitary b⟩
theorem conjugate_eq_diagonal (A : Matrix ι ι ℂ)
(b : OrthonormalBasis ι ℂ (EuclideanSpace ℂ ι)) (d : ι → ℂ)
(hd : ∀ i, A.toEuclideanLin (b i) = d i • b i) :
A = (basisUnitary b : Matrix ι ι ℂ) * Matrix.diagonal d * star (basisUnitary b : Matrix ι ι ℂ) := by
let U := basisUnitary b
have hAU : A*(U : Matrix ι ι ℂ) = (U : Matrix ι ι ℂ)*Matrix.diagonal d := by
ext i j
have h := congrArg (fun v : EuclideanSpace ℂ ι => v i) (hd j)
rw [Matrix.mul_diagonal]
change (∑ k, A i k * (b j) k) = (b j) i * d j
simpa [Matrix.toLpLin_apply,Matrix.mulVec,dotProduct,smul_eq_mul,mul_comm] using h
have hright := congrArg (fun M : Matrix ι ι ℂ => M*star (U : Matrix ι ι ℂ)) hAU
have hu : (U : Matrix ι ι ℂ)*star (U : Matrix ι ι ℂ) = 1 := Unitary.coe_mul_star_self U
rw [mul_assoc,hu,mul_one] at hright
exact hright
theorem exists_unitary (A B : Matrix ι ι ℂ) (hA : A.IsHermitian)
(hB : B.IsHermitian) (hAB : Commute A B) :
∃ (U : unitary (Matrix ι ι ℂ)) (d e : ι → ℂ),
A = (U : Matrix ι ι ℂ)*Matrix.diagonal d*star (U : Matrix ι ι ℂ) ∧
B = (U : Matrix ι ι ℂ)*Matrix.diagonal e*star (U : Matrix ι ι ℂ) := by
obtain ⟨b,d,e,hd,he⟩ := exists_basis A B hA hB hAB
exact ⟨basisUnitary b,d,e,conjugate_eq_diagonal A b d hd,conjugate_eq_diagonal B b e he⟩
open scoped ComplexOrder
theorem diagonal_nonneg (A : Matrix ι ι ℂ) (hA : A.PosSemidef)
(U : unitary (Matrix ι ι ℂ)) (d : ι → ℂ)
(hd : A = (U : Matrix ι ι ℂ)*Matrix.diagonal d*star (U : Matrix ι ι ℂ)) :
∀ i, 0 ≤ d i := by
have hp := hA.conjTranspose_mul_mul_same (U : Matrix ι ι ℂ)
rw [hd] at hp
have hu : star (U : Matrix ι ι ℂ)*(U : Matrix ι ι ℂ)=1 := Unitary.coe_star_mul_self U
rw [← Matrix.star_eq_conjTranspose,← mul_assoc,← mul_assoc,hu,one_mul,
mul_assoc,hu,mul_one] at hp
exact Matrix.posSemidef_diagonal_iff.mp hp
theorem exists_positive_unitary (A B : Matrix ι ι ℂ) (hA : A.PosSemidef)
(hB : B.PosSemidef) (hAB : Commute A B) :
∃ (U : unitary (Matrix ι ι ℂ)) (d e : ι → ℝ),
(∀ i, 0 ≤ d i) ∧ (∀ i, 0 ≤ e i) ∧
A = (U : Matrix ι ι ℂ)*Matrix.diagonal (fun i => (d i:ℂ))*star (U : Matrix ι ι ℂ) ∧
B = (U : Matrix ι ι ℂ)*Matrix.diagonal (fun i => (e i:ℂ))*star (U : Matrix ι ι ℂ) := by
obtain ⟨U,d,e,hd,he⟩ := exists_unitary A B hA.isHermitian hB.isHermitian hAB
have hd0 := diagonal_nonneg A hA U d hd
have he0 := diagonal_nonneg B hB U e he
have hdr (i : ι) : ((d i).re:ℂ) = d i := by
exact Complex.ext rfl (by simpa using (Complex.nonneg_iff.mp (hd0 i)).2)
have her (i : ι) : ((e i).re:ℂ) = e i := by
exact Complex.ext rfl (by simpa using (Complex.nonneg_iff.mp (he0 i)).2)
refine ⟨U,(fun i => (d i).re),(fun i => (e i).re),
(fun i => (Complex.nonneg_iff.mp (hd0 i)).1),
(fun i => (Complex.nonneg_iff.mp (he0 i)).1),?_,?_⟩
· simpa only [hdr] using hd
· simpa only [her] using he
end PolynomialPEPS.Subvolume.JointDiagonalization
end
end OAI
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