The General Linear group $GL(n, R)$

This file defines the elements of the General Linear group Matrix.GeneralLinearGroup n R, consisting of all invertible n by n R-matrices.

Main definitions

• Matrix.GeneralLinearGroup is the type of matrices over R which are units in the matrix ring.
• Matrix.GLPos gives the subgroup of matrices with positive determinant (over a linear ordered ring).

Tags

matrix group, group, matrix inverse

@[reducible, inline]

abbrev Matrix.GeneralLinearGroup (n : Type u) (R : Type v) [DecidableEq n] [Fintype n] [Semiring R] : Type (max v u)

GL n R is the group of n by n R-matrices with unit determinant. Defined as a subtype of matrices

def Matrix.termGL : Lean.ParserDescr

GL n R is the group of n by n R-matrices with unit determinant. Defined as a subtype of matrices

def Matrix.GeneralLinearGroup.scalar (n : Type u) [DecidableEq n] [Fintype n] {R : Type v} [Semiring R] : Rˣ →* GL n R

Scalar matrix as an element of GL n R.

@[simp]

theorem Matrix.GeneralLinearGroup.val_scalar_apply (n : Type u) [DecidableEq n] [Fintype n] {R : Type v} [Semiring R] (u : Rˣ) : ↑((scalar n) u) = diagonal fun (x : n) => ↑u

@[simp]

theorem Matrix.GeneralLinearGroup.val_inv_scalar_apply (n : Type u) [DecidableEq n] [Fintype n] {R : Type v} [Semiring R] (u : Rˣ) : ↑((scalar n) u)⁻¹ = diagonal fun (x : n) => ↑u⁻¹

@[implicit_reducible]

instance Matrix.GeneralLinearGroup.instCoeFun {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [Semiring R] : CoeFun (GL n R) fun (x : GL n R) => n → n → R

def Matrix.GeneralLinearGroup.det {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] : GL n R →* Rˣ

The determinant of a unit matrix is itself a unit.

@[simp]

theorem Matrix.GeneralLinearGroup.val_inv_det_apply {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] (A : GL n R) : ↑(det A)⁻¹ = (↑A⁻¹).det

@[simp]

theorem Matrix.GeneralLinearGroup.val_det_apply {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] (A : GL n R) : ↑(det A) = (↑A).det

theorem Matrix.GeneralLinearGroup.det_ne_zero {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] [Nontrivial R] (g : GL n R) : (↑g).det ≠ 0

@[simp]

theorem Matrix.GeneralLinearGroup.det_scalar {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] (u : Rˣ) : det ((scalar n) u) = u ^ Fintype.card n

def Matrix.GeneralLinearGroup.toLin {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] : GL n R ≃* LinearMap.GeneralLinearGroup R (n → R)

The groups GL n R (notation for Matrix.GeneralLinearGroup n R) and LinearMap.GeneralLinearGroup R (n → R) are multiplicatively equivalent

noncomputable def Matrix.GeneralLinearGroup.toLin' {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] {V : Type u_1} [AddCommGroup V] [Module R V] (b : Module.Basis n R V) : GL n R ≃* LinearMap.GeneralLinearGroup R V

The isomorphism from GL n R to the general linear group of a module associated with a basis.

theorem Matrix.GeneralLinearGroup.toLin'_apply {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] {V : Type u_1} [AddCommGroup V] [Module R V] (b : Module.Basis n R V) (M : GL n R) (v : V) : ((toLin' b) M).toLinearEquiv v = (Fintype.linearCombination R ⇑b) ((↑M).mulVec ⇑(b.repr v))

def Matrix.GeneralLinearGroup.mk' {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] (A : Matrix n n R) : Invertible A.det → GL n R

Given a matrix with invertible determinant, we get an element of GL n R.

@[simp]

theorem Matrix.GeneralLinearGroup.val_mk' {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] (A : Matrix n n R) (x✝ : Invertible A.det) : ↑(mk' A x✝) = A

noncomputable def Matrix.GeneralLinearGroup.mk'' {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] (A : Matrix n n R) (h : IsUnit A.det) : GL n R

Given a matrix with unit determinant, we get an element of GL n R.

@[simp]

theorem Matrix.GeneralLinearGroup.val_mk'' {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] (A : Matrix n n R) (h : IsUnit A.det) : ↑(mk'' A h) = A

def Matrix.GeneralLinearGroup.mkOfDetNeZero {n : Type u} [DecidableEq n] [Fintype n] {K : Type u_1} [Field K] (A : Matrix n n K) (h : A.det ≠ 0) : GL n K

Given a matrix with non-zero determinant over a field, we get an element of GL n K.

@[simp]

theorem Matrix.GeneralLinearGroup.val_mkOfDetNeZero {n : Type u} [DecidableEq n] [Fintype n] {K : Type u_1} [Field K] (A : Matrix n n K) (h : A.det ≠ 0) : ↑(mkOfDetNeZero A h) = A

theorem Matrix.GeneralLinearGroup.ext_iff {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] (A B : GL n R) : A = B ↔ ∀ (i j : n), ↑A i j = ↑B i j

theorem Matrix.GeneralLinearGroup.ext {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] ⦃A B : GL n R⦄ (h : ∀ (i j : n), ↑A i j = ↑B i j) : A = B

Not marked @[ext] as the ext tactic already solves this.

@[simp]

theorem Matrix.GeneralLinearGroup.coe_mul {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] (A B : GL n R) : ↑(A * B) = ↑A * ↑B

@[simp]

theorem Matrix.GeneralLinearGroup.coe_one {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] : ↑1 = 1

theorem Matrix.GeneralLinearGroup.coe_inv {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] (A : GL n R) : ↑A⁻¹ = (↑A)⁻¹

@[simp]

theorem Matrix.GeneralLinearGroup.coe_toLin {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] (A : GL n R) : ↑(toLin A) = (↑A).mulVecLin

@[simp]

theorem Matrix.GeneralLinearGroup.toLin_apply {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] (A : GL n R) (v : n → R) : ↑(toLin A) v = (↑A).mulVecLin v

def Matrix.GeneralLinearGroup.map {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] {S : Type u_1} [CommRing S] (f : R →+* S) : GL n R →* GL n S

A ring homomorphism f : R →+* S induces a homomorphism GLₙ(f) : GLₙ(R) →* GLₙ(S).

@[simp]

theorem Matrix.GeneralLinearGroup.val_map_apply {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] {S : Type u_1} [CommRing S] (f : R →+* S) (u : (Matrix n n R)ˣ) : ↑((map f) u) = (↑u).map ⇑f

@[simp]

theorem Matrix.GeneralLinearGroup.map_id {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] : map (RingHom.id R) = MonoidHom.id (GL n R)

@[simp]

theorem Matrix.GeneralLinearGroup.map_apply {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] {S : Type u_1} [CommRing S] (f : R →+* S) (i j : n) (g : GL n R) : ↑((map f) g) i j = f (↑g i j)

@[simp]

theorem Matrix.GeneralLinearGroup.map_comp {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] {S : Type u_1} {T : Type u_2} [CommRing S] [CommRing T] (f : T →+* R) (g : R →+* S) : map (g.comp f) = (map g).comp (map f)

@[simp]

theorem Matrix.GeneralLinearGroup.map_comp_apply {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] {S : Type u_1} {T : Type u_2} [CommRing S] [CommRing T] (f : T →+* R) (g : R →+* S) (x : GL n T) : ((map g).comp (map f)) x = (map g) ((map f) x)

@[simp]

theorem Matrix.GeneralLinearGroup.map_one {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] {S : Type u_1} [CommRing S] (f : R →+* S) : (map f) 1 = 1

theorem Matrix.GeneralLinearGroup.map_mul {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] {S : Type u_1} [CommRing S] (f : R →+* S) (g h : GL n R) : (map f) (g * h) = (map f) g * (map f) h

theorem Matrix.GeneralLinearGroup.map_inv {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] {S : Type u_1} [CommRing S] (f : R →+* S) (g : GL n R) : (map f) g⁻¹ = ((map f) g)⁻¹

theorem Matrix.GeneralLinearGroup.map_det {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] {S : Type u_1} [CommRing S] (f : R →+* S) (g : GL n R) : det ((map f) g) = (Units.map ↑f) (det g)

theorem Matrix.GeneralLinearGroup.map_mul_map_inv {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] {S : Type u_1} [CommRing S] (f : R →+* S) (g : GL n R) : (map f) g * (map f) g⁻¹ = 1

theorem Matrix.GeneralLinearGroup.map_inv_mul_map {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] {S : Type u_1} [CommRing S] (f : R →+* S) (g : GL n R) : (map f) g⁻¹ * (map f) g = 1

@[simp]

theorem Matrix.GeneralLinearGroup.coe_map_mul_map_inv {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] {S : Type u_1} [CommRing S] (f : R →+* S) (g : GL n R) : (↑g).map ⇑f * (↑g)⁻¹.map ⇑f = 1

@[simp]

theorem Matrix.GeneralLinearGroup.coe_map_inv_mul_map {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] {S : Type u_1} [CommRing S] (f : R →+* S) (g : GL n R) : (↑g)⁻¹.map ⇑f * (↑g).map ⇑f = 1

def Matrix.GeneralLinearGroup.kronecker {n : Type u} [DecidableEq n] [Fintype n] {R : Type u_3} {m : Type u_4} [CommSemiring R] [Fintype m] [DecidableEq m] (x : GL n R) (y : GL m R) : GL (n × m) R

The invertible kronecker matrix of invertible matrices.

theorem Matrix.IsUnit.kronecker {n : Type u} [DecidableEq n] [Fintype n] {R : Type u_3} {m : Type u_4} [CommSemiring R] [Fintype m] [DecidableEq m] {x : Matrix n n R} {y : Matrix m m R} (hx : IsUnit x) (hy : IsUnit y) : IsUnit (kroneckerMap (fun (x1 x2 : R) => x1 * x2) x y)

def Matrix.SpecialLinearGroup.toGL {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] : SpecialLinearGroup n R →* GL n R

toGL is the map from the special linear group to the general linear group.

@[implicit_reducible]

instance Matrix.SpecialLinearGroup.hasCoeToGeneralLinearGroup {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] : Coe (SpecialLinearGroup n R) (GL n R)

theorem Matrix.SpecialLinearGroup.toGL_injective {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] : Function.Injective ⇑toGL

@[simp]

theorem Matrix.SpecialLinearGroup.toGL_inj {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] (g g' : SpecialLinearGroup n R) : toGL g = toGL g' ↔ g = g'

@[simp]

theorem Matrix.SpecialLinearGroup.coeToGL_det {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] (g : SpecialLinearGroup n R) : GeneralLinearGroup.det (toGL g) = 1

@[simp]

theorem Matrix.SpecialLinearGroup.coe_GL_coe_matrix {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] (g : SpecialLinearGroup n R) : ↑(toGL g) = ↑g

def Matrix.SpecialLinearGroup.mapGL {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] (S : Type u_1) [CommRing S] [Algebra R S] : SpecialLinearGroup n R →* GL n S

mapGL is the map from the special linear group over R to the general linear group over S, where S is an R-algebra.

@[simp]

theorem Matrix.SpecialLinearGroup.mapGL_inj {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] {S : Type u_1} [CommRing S] [Algebra R S] [FaithfulSMul R S] (g g' : SpecialLinearGroup n R) : (mapGL S) g = (mapGL S) g' ↔ g = g'

theorem Matrix.SpecialLinearGroup.mapGL_injective {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] {S : Type u_1} [CommRing S] [Algebra R S] [FaithfulSMul R S] : Function.Injective ⇑(mapGL S)

@[simp]

theorem Matrix.SpecialLinearGroup.mapGL_coe_matrix {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] {S : Type u_1} [CommRing S] [Algebra R S] (g : SpecialLinearGroup n R) : ↑((mapGL S) g) = ↑((map (algebraMap R S)) g)

@[simp]

theorem Matrix.SpecialLinearGroup.map_mapGL {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] {S : Type u_1} [CommRing S] [Algebra R S] {T : Type u_2} [CommRing T] [Algebra R T] [Algebra S T] [IsScalarTower R S T] (g : SpecialLinearGroup n R) : (GeneralLinearGroup.map (algebraMap S T)) ((mapGL S) g) = (mapGL T) g

@[simp]

theorem Matrix.SpecialLinearGroup.det_mapGL {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] {S : Type u_1} [CommRing S] [Algebra R S] (g : SpecialLinearGroup n R) : GeneralLinearGroup.det ((mapGL S) g) = 1

def Matrix.GLPos (n : Type u) (R : Type v) [DecidableEq n] [Fintype n] [CommRing R] [LinearOrder R] [IsStrictOrderedRing R] : Subgroup (GL n R)

This is the subgroup of nxn matrices with entries over a linear ordered ring and positive determinant.

def MatrixGroups.«termGL(_,_)⁺» : Lean.ParserDescr

This is the subgroup of nxn matrices with entries over a linear ordered ring and positive determinant.

@[simp]

theorem Matrix.mem_glpos {n : Type u} {R : Type v} [DecidableEq n] [Fintype n] [CommRing R] [LinearOrder R] [IsStrictOrderedRing R] (A : GL n R) : A ∈ GLPos n R ↔ 0 < ↑(GeneralLinearGroup.det A)

theorem Matrix.GLPos.det_ne_zero {n : Type u} {R : Type v} [DecidableEq n] [Fintype n] [CommRing R] [LinearOrder R] [IsStrictOrderedRing R] (A : ↥(GLPos n R)) : (↑↑A).det ≠ 0

@[implicit_reducible]

instance Matrix.instNegSubtypeGeneralLinearGroupMemSubgroupGLPos {n : Type u} {R : Type v} [DecidableEq n] [Fintype n] [CommRing R] [LinearOrder R] [IsStrictOrderedRing R] [Fact (Even (Fintype.card n))] : Neg ↥(GLPos n R)

Formal operation of negation on general linear group on even cardinality n given by negating each element.

@[simp]

theorem Matrix.GLPos.coe_neg_GL {n : Type u} {R : Type v} [DecidableEq n] [Fintype n] [CommRing R] [LinearOrder R] [IsStrictOrderedRing R] [Fact (Even (Fintype.card n))] (g : ↥(GLPos n R)) : ↑(-g) = -↑g

@[simp]

theorem Matrix.GLPos.coe_neg {n : Type u} {R : Type v} [DecidableEq n] [Fintype n] [CommRing R] [LinearOrder R] [IsStrictOrderedRing R] [Fact (Even (Fintype.card n))] (g : ↥(GLPos n R)) : ↑↑(-g) = -↑↑g

@[simp]

theorem Matrix.GLPos.coe_neg_apply {n : Type u} {R : Type v} [DecidableEq n] [Fintype n] [CommRing R] [LinearOrder R] [IsStrictOrderedRing R] [Fact (Even (Fintype.card n))] (g : ↥(GLPos n R)) (i j : n) : ↑↑(-g) i j = -↑↑g i j

@[implicit_reducible]

instance Matrix.instHasDistribNegSubtypeGeneralLinearGroupMemSubgroupGLPos {n : Type u} {R : Type v} [DecidableEq n] [Fintype n] [CommRing R] [LinearOrder R] [IsStrictOrderedRing R] [Fact (Even (Fintype.card n))] : HasDistribNeg ↥(GLPos n R)

def Matrix.SpecialLinearGroup.toGLPos {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] [LinearOrder R] [IsStrictOrderedRing R] : SpecialLinearGroup n R →* ↥(GLPos n R)

Matrix.SpecialLinearGroup n R embeds into GL_pos n R

@[implicit_reducible]

instance Matrix.SpecialLinearGroup.instCoeSubtypeGeneralLinearGroupMemSubgroupGLPos {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] [LinearOrder R] [IsStrictOrderedRing R] : Coe (SpecialLinearGroup n R) ↥(GLPos n R)

theorem Matrix.SpecialLinearGroup.toGLPos_injective {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] [LinearOrder R] [IsStrictOrderedRing R] : Function.Injective ⇑toGLPos

@[simp]

theorem Matrix.SpecialLinearGroup.coe_GLPos_coe_GL_coe_matrix {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] [LinearOrder R] [IsStrictOrderedRing R] (g : SpecialLinearGroup n R) : ↑↑(toGLPos g) = ↑g

Coercing a Matrix.SpecialLinearGroup via GL_pos and GL is the same as coercing straight to a matrix.

@[simp]

theorem Matrix.SpecialLinearGroup.coe_to_GLPos_to_GL_det {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] [LinearOrder R] [IsStrictOrderedRing R] (g : SpecialLinearGroup n R) : GeneralLinearGroup.det ↑(toGLPos g) = 1

theorem Matrix.SpecialLinearGroup.coe_GLPos_neg {n : Type u} [DecidableEq n] [Fintype n] {R : Type v} [CommRing R] [LinearOrder R] [IsStrictOrderedRing R] [Fact (Even (Fintype.card n))] (g : SpecialLinearGroup n R) : toGLPos (-g) = -toGLPos g