HepLean Documentation

Mathlib.LinearAlgebra.UnitaryGroup

The Unitary Group #

This file defines elements of the unitary group Matrix.unitaryGroup n α, where α is a StarRing. This consists of all n by n matrices with entries in α such that the star-transpose is its inverse. In addition, we define the group structure on Matrix.unitaryGroup n α, and the embedding into the general linear group LinearMap.GeneralLinearGroup α (n → α).

We also define the orthogonal group Matrix.orthogonalGroup n β, where β is a CommRing.

Main Definitions #

References #

Tags #

matrix group, group, unitary group, orthogonal group

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@[reducible, inline]
abbrev Matrix.unitaryGroup (n : Type u) [DecidableEq n] [Fintype n] (α : Type v) [CommRing α] [StarRing α] :
Submonoid (Matrix n n α)

Matrix.unitaryGroup n is the group of n by n matrices where the star-transpose is the inverse.

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    theorem Matrix.mem_unitaryGroup_iff {n : Type u} [DecidableEq n] [Fintype n] {α : Type v} [CommRing α] [StarRing α] {A : Matrix n n α} :
    A ∈ Matrix.unitaryGroup n α ↔ A * star A = 1
    source
    theorem Matrix.mem_unitaryGroup_iff' {n : Type u} [DecidableEq n] [Fintype n] {α : Type v} [CommRing α] [StarRing α] {A : Matrix n n α} :
    A ∈ Matrix.unitaryGroup n α ↔ star A * A = 1
    source
    theorem Matrix.det_of_mem_unitary {n : Type u} [DecidableEq n] [Fintype n] {α : Type v} [CommRing α] [StarRing α] {A : Matrix n n α} (hA : A ∈ Matrix.unitaryGroup n α) :
    A.det ∈ unitary α
    source
    instance Matrix.UnitaryGroup.coeMatrix {n : Type u} [DecidableEq n] [Fintype n] {α : Type v} [CommRing α] [StarRing α] :
    Coe (↥(Matrix.unitaryGroup n α)) (Matrix n n α)
    Equations
    • Matrix.UnitaryGroup.coeMatrix = { coe := Subtype.val }
    source
    instance Matrix.UnitaryGroup.coeFun {n : Type u} [DecidableEq n] [Fintype n] {α : Type v} [CommRing α] [StarRing α] :
    CoeFun ↥(Matrix.unitaryGroup n α) fun (x : ↥(Matrix.unitaryGroup n α)) => n → n → α
    Equations
    source
    def Matrix.UnitaryGroup.toLin' {n : Type u} [DecidableEq n] [Fintype n] {α : Type v} [CommRing α] [StarRing α] (A : ↥(Matrix.unitaryGroup n α)) :
    (n → α) →ₗ[α] n → α

    Matrix.UnitaryGroup.toLin' A is matrix multiplication of vectors by A, as a linear map.

    After the group structure on Matrix.unitaryGroup n is defined, we show in Matrix.UnitaryGroup.toLinearEquiv that this gives a linear equivalence.

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      theorem Matrix.UnitaryGroup.ext_iff {n : Type u} [DecidableEq n] [Fintype n] {α : Type v} [CommRing α] [StarRing α] (A : ↥(Matrix.unitaryGroup n α)) (B : ↥(Matrix.unitaryGroup n α)) :
      A = B ↔ ∀ (i j : n), ↑A i j = ↑B i j
      source
      theorem Matrix.UnitaryGroup.ext {n : Type u} [DecidableEq n] [Fintype n] {α : Type v} [CommRing α] [StarRing α] (A : ↥(Matrix.unitaryGroup n α)) (B : ↥(Matrix.unitaryGroup n α)) :
      (∀ (i j : n), ↑A i j = ↑B i j) → A = B
      source
      theorem Matrix.UnitaryGroup.star_mul_self {n : Type u} [DecidableEq n] [Fintype n] {α : Type v} [CommRing α] [StarRing α] (A : ↥(Matrix.unitaryGroup n α)) :
      star ↑A * ↑A = 1
      source
      @[simp]
      theorem Matrix.UnitaryGroup.det_isUnit {n : Type u} [DecidableEq n] [Fintype n] {α : Type v} [CommRing α] [StarRing α] (A : ↥(Matrix.unitaryGroup n α)) :
      IsUnit (↑A).det
      source
      @[simp]
      theorem Matrix.UnitaryGroup.inv_val {n : Type u} [DecidableEq n] [Fintype n] {α : Type v} [CommRing α] [StarRing α] (A : ↥(Matrix.unitaryGroup n α)) :
      ↑A⁻¹ = star ↑A
      source
      @[simp]
      theorem Matrix.UnitaryGroup.inv_apply {n : Type u} [DecidableEq n] [Fintype n] {α : Type v} [CommRing α] [StarRing α] (A : ↥(Matrix.unitaryGroup n α)) :
      ↑A⁻¹ = star ↑A
      source
      @[simp]
      theorem Matrix.UnitaryGroup.mul_val {n : Type u} [DecidableEq n] [Fintype n] {α : Type v} [CommRing α] [StarRing α] (A : ↥(Matrix.unitaryGroup n α)) (B : ↥(Matrix.unitaryGroup n α)) :
      ↑(A * B) = ↑A * ↑B
      source
      @[simp]
      theorem Matrix.UnitaryGroup.mul_apply {n : Type u} [DecidableEq n] [Fintype n] {α : Type v} [CommRing α] [StarRing α] (A : ↥(Matrix.unitaryGroup n α)) (B : ↥(Matrix.unitaryGroup n α)) :
      ↑(A * B) = ↑A * ↑B
      source
      @[simp]
      theorem Matrix.UnitaryGroup.one_val {n : Type u} [DecidableEq n] [Fintype n] {α : Type v} [CommRing α] [StarRing α] :
      ↑1 = 1
      source
      @[simp]
      theorem Matrix.UnitaryGroup.one_apply {n : Type u} [DecidableEq n] [Fintype n] {α : Type v} [CommRing α] [StarRing α] :
      ↑1 = 1
      source
      @[simp]
      theorem Matrix.UnitaryGroup.toLin'_mul {n : Type u} [DecidableEq n] [Fintype n] {α : Type v} [CommRing α] [StarRing α] (A : ↥(Matrix.unitaryGroup n α)) (B : ↥(Matrix.unitaryGroup n α)) :
      Matrix.UnitaryGroup.toLin' (A * B) = Matrix.UnitaryGroup.toLin' A ∘ₗ Matrix.UnitaryGroup.toLin' B
      source
      @[simp]
      theorem Matrix.UnitaryGroup.toLin'_one {n : Type u} [DecidableEq n] [Fintype n] {α : Type v} [CommRing α] [StarRing α] :
      Matrix.UnitaryGroup.toLin' 1 = LinearMap.id
      source
      def Matrix.UnitaryGroup.toLinearEquiv {n : Type u} [DecidableEq n] [Fintype n] {α : Type v} [CommRing α] [StarRing α] (A : ↥(Matrix.unitaryGroup n α)) :
      (n → α) ≃ₗ[α] n → α

      Matrix.unitaryGroup.toLinearEquiv A is matrix multiplication of vectors by A, as a linear equivalence.

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        def Matrix.UnitaryGroup.toGL {n : Type u} [DecidableEq n] [Fintype n] {α : Type v} [CommRing α] [StarRing α] (A : ↥(Matrix.unitaryGroup n α)) :
        LinearMap.GeneralLinearGroup α (n → α)

        Matrix.unitaryGroup.toGL is the map from the unitary group to the general linear group

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          theorem Matrix.UnitaryGroup.coe_toGL {n : Type u} [DecidableEq n] [Fintype n] {α : Type v} [CommRing α] [StarRing α] (A : ↥(Matrix.unitaryGroup n α)) :
          ↑(Matrix.UnitaryGroup.toGL A) = Matrix.UnitaryGroup.toLin' A
          source
          @[simp]
          theorem Matrix.UnitaryGroup.toGL_one {n : Type u} [DecidableEq n] [Fintype n] {α : Type v} [CommRing α] [StarRing α] :
          Matrix.UnitaryGroup.toGL 1 = 1
          source
          @[simp]
          theorem Matrix.UnitaryGroup.toGL_mul {n : Type u} [DecidableEq n] [Fintype n] {α : Type v} [CommRing α] [StarRing α] (A : ↥(Matrix.unitaryGroup n α)) (B : ↥(Matrix.unitaryGroup n α)) :
          Matrix.UnitaryGroup.toGL (A * B) = Matrix.UnitaryGroup.toGL A * Matrix.UnitaryGroup.toGL B
          source
          def Matrix.UnitaryGroup.embeddingGL {n : Type u} [DecidableEq n] [Fintype n] {α : Type v} [CommRing α] [StarRing α] :
          ↥(Matrix.unitaryGroup n α) →* LinearMap.GeneralLinearGroup α (n → α)

          Matrix.unitaryGroup.embeddingGL is the embedding from Matrix.unitaryGroup n α to LinearMap.GeneralLinearGroup n α.

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            @[reducible, inline]
            abbrev Matrix.specialUnitaryGroup (n : Type u) [DecidableEq n] [Fintype n] (α : Type v) [CommRing α] [StarRing α] :
            Submonoid (Matrix n n α)

            Matrix.specialUnitaryGroup is the group of unitary n by n matrices where the determinant is 1. (This definition is only correct if 2 is invertible.)

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              theorem Matrix.mem_specialUnitaryGroup_iff {n : Type u} [DecidableEq n] [Fintype n] {α : Type v} [CommRing α] [StarRing α] {A : Matrix n n α} :
              A ∈ Matrix.specialUnitaryGroup n α ↔ A ∈ Matrix.unitaryGroup n α ∧ A.det = 1
              source
              @[reducible, inline]
              abbrev Matrix.orthogonalGroup (n : Type u) [DecidableEq n] [Fintype n] (β : Type v) [CommRing β] :
              Submonoid (Matrix n n β)

              Matrix.orthogonalGroup n is the group of n by n matrices where the transpose is the inverse.

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                theorem Matrix.mem_orthogonalGroup_iff (n : Type u) [DecidableEq n] [Fintype n] (β : Type v) [CommRing β] {A : Matrix n n β} :
                A ∈ Matrix.orthogonalGroup n β ↔ A * star A = 1
                source
                theorem Matrix.mem_orthogonalGroup_iff' (n : Type u) [DecidableEq n] [Fintype n] (β : Type v) [CommRing β] {A : Matrix n n β} :
                A ∈ Matrix.orthogonalGroup n β ↔ star A * A = 1
                source
                @[reducible, inline]
                abbrev Matrix.specialOrthogonalGroup (n : Type u) [DecidableEq n] [Fintype n] (β : Type v) [CommRing β] :
                Submonoid (Matrix n n β)

                Matrix.specialOrthogonalGroup n is the group of orthogonal n by n where the determinant is one. (This definition is only correct if 2 is invertible.)

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                  theorem Matrix.mem_specialOrthogonalGroup_iff {n : Type u} [DecidableEq n] [Fintype n] {β : Type v} [CommRing β] {A : Matrix n n β} :
                  A ∈ Matrix.specialOrthogonalGroup n β ↔ A ∈ Matrix.orthogonalGroup n β ∧ A.det = 1