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{{Group theory sidebar |top/lie}}{{Use American English|date = January 2019}}
{{Short description|n x n invertible matrices over a ring}}
{{Lie groups |classical}}

In [[mathematics]], the '''general linear group''' of degree ''n'' is the set of {{nowrap|''n''×''n''}} [[invertible matrix|invertible matrices]], together with the operation of ordinary [[matrix multiplication]]. This forms a [[group (mathematics)|group]], because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible, with identity matrix as the identity element of the group. The group is so named because the columns of an invertible matrix are [[linearly independent]], hence the vectors/points they define are in [[general linear position]], and matrices in the general linear group take points in general linear position to points in general linear position.

To be more precise, it is necessary to specify what kind of objects may appear in the entries of the matrix. For example, the general linear group over '''R''' (the set of [[real numbers]]) is the group of {{nowrap|''n''×''n''}} invertible matrices of real numbers, and is denoted by GL<sub>''n''</sub>('''R''') or {{nowrap|GL(''n'', '''R''')}}.

More generally, the general linear group of degree ''n'' over any [[field (mathematics)|field]] ''F'' (such as the [[complex number]]s), or a [[ring (mathematics)|ring]] ''R'' (such as the ring of [[integer]]s), is the set of {{nowrap|''n''×''n''}} invertible matrices with entries from ''F'' (or ''R''), again with matrix multiplication as the group operation.<ref name="ring">Here rings are assumed to be [[Ring (mathematics)#Notes on the definition|associative and unital]].</ref> Typical notation is GL<sub>''n''</sub>(''F'') or {{nowrap|GL(''n'', ''F'')}}, or simply GL(''n'') if the field is understood.

More generally still, the [[#General linear group of a vector space|general linear group of a vector space]] GL(''V'') is the abstract [[automorphism group]], not necessarily written as matrices.

The '''[[#Special linear group|special linear group]]''', written {{nowrap|SL(''n'', ''F'')}} or SL<sub>''n''</sub>(''F''), is the [[subgroup]] of {{nowrap|GL(''n'', ''F'')}} consisting of matrices with a [[determinant]] of 1.

The group {{nowrap|GL(''n'', ''F'')}} and its [[subgroup]]s are often called '''linear groups''' or '''matrix groups''' (the abstract group GL(''V'') is a linear group but not a matrix group). These groups are important in the theory of [[group representation]]s, and also arise in the study of spatial [[symmetry|symmetries]] and symmetries of [[vector space]]s in general, as well as the study of [[polynomials]]. The [[modular group]] may be realised as a quotient of the special linear group {{nowrap|SL(2, '''Z''')}}.

If {{nowrap|''n'' ≥ 2}}, then the group {{nowrap|GL(''n'', ''F'')}} is not [[abelian group|abelian]].

== General linear group of a vector space ==

If ''V'' is a [[vector space]] over the field ''F'', the general linear group of ''V'', written GL(''V'') or Aut(''V''), is the group of all [[automorphism]]s of ''V'', i.e. the set of all [[bijective]] [[linear transformation]]s {{nowrap|''V'' → ''V''}}, together with functional composition as group operation. If ''V'' has finite [[Hamel dimension|dimension]] ''n'', then GL(''V'') and {{nowrap|GL(''n'', ''F'')}} are [[group isomorphism|isomorphic]]. The isomorphism is not canonical; it depends on a choice of [[basis (linear algebra)|basis]] in ''V''. Given a basis {{nowrap|(''e''<sub>1</sub>, ..., ''e''<sub>''n''</sub>)}} of ''V'' and an automorphism ''T'' in GL(''V''), we have then for every basis vector ''e''<sub>''i''</sub> that
: <math>Te_i = \sum_{j=1}^n a_{ij} e_j</math>
for some constants ''a''<sub>''ij''</sub> in ''F''; the matrix corresponding to ''T'' is then just the matrix with entries given by the ''a''<sub>''ij''</sub>.

In a similar way, for a commutative ring ''R'' the group {{nowrap|GL(''n'', ''R'')}} may be interpreted as the group of automorphisms of a ''[[free module|free]]'' ''R''-module ''M'' of rank ''n''. One can also define GL(''M'') for any ''R''-module, but in general this is not isomorphic to {{nowrap|GL(''n'', ''R'')}} (for any ''n'').

== In terms of determinants ==

Over a field ''F'', a matrix is [[invertible]] if and only if its [[determinant]] is nonzero. Therefore, an alternative definition of {{nowrap|GL(''n'', ''F'')}} is as the group of matrices with nonzero determinant.

Over a [[commutative ring]] ''R'', more care is needed: a matrix over ''R'' is invertible if and only if its determinant is a [[unit (ring theory)|unit]] in ''R'', that is, if its determinant is invertible in ''R''. Therefore, {{nowrap|GL(''n'', ''R'')}} may be defined as the group of matrices whose determinants are units.

Over a non-commutative ring ''R'', determinants are not at all well behaved. In this case, {{nowrap|GL(''n'', ''R'')}} may be defined as the [[unit group]] of the [[matrix ring]] {{nowrap|M(''n'', ''R'')}}.

== As a Lie group ==

=== Real case ===

The general linear group {{nowrap|GL(''n'', '''R''')}} over the field of [[real number]]s is a real [[Lie group]] of dimension ''n''<sup>2</sup>. To see this, note that the set of all {{nowrap|''n''×''n''}} real matrices, M<sub>''n''</sub>('''R'''), forms a [[real vector space]] of dimension ''n''<sup>2</sup>. The subset {{nowrap|GL(''n'', '''R''')}} consists of those matrices whose [[determinant]] is non-zero. The determinant is a [[polynomial]] map, and hence {{nowrap|GL(''n'', '''R''')}} is an [[algebraic variety|open affine subvariety]] of M<sub>''n''</sub>('''R''') (a [[non-empty]] [[open subset]] of M<sub>''n''</sub>('''R''') in the [[Zariski topology]]), and therefore<ref>
Since the Zariski topology is [[coarsest topology|coarser]] than the metric topology; equivalently, polynomial maps are [[continuous function (topology)|continuous]].</ref>
a [[smooth manifold]] of the same dimension.

The [[Lie algebra]] of {{nowrap|GL(''n'', '''R''')}}, denoted <math>\mathfrak{gl}_n,</math> consists of all {{nowrap|''n''×''n''}} real matrices with the [[commutator]] serving as the Lie bracket.

As a manifold, {{nowrap|GL(''n'', '''R''')}} is not [[connected space|connected]] but rather has two [[connected space|connected components]]: the matrices with positive determinant and the ones with negative determinant. The [[identity component]], denoted by {{nowrap|GL<sup>+</sup>(''n'', '''R''')}}, consists of the real {{nowrap|''n''×''n''}} matrices with positive determinant. This is also a Lie group of dimension ''n''<sup>2</sup>; it has the same Lie algebra as {{nowrap|GL(''n'', '''R''')}}.

The group {{nowrap|GL(''n'', '''R''')}} is also [[compact space|noncompact]]. "The"<ref>A maximal compact subgroup is not unique, but is [[essentially unique]], hence one often refers to "the" maximal compact subgroup.</ref> [[maximal compact subgroup]] of {{nowrap|GL(''n'', '''R''')}} is the [[orthogonal group]] O(''n''), while "the" maximal compact subgroup of {{nowrap|GL<sup>+</sup>(''n'', '''R''')}} is the [[special orthogonal group]] SO(''n''). As for SO(''n''), the group {{nowrap|GL<sup>+</sup>(''n'', '''R''')}} is not [[simply connected]] (except when {{nowrap|1=''n'' = 1)}}, but rather has a [[fundamental group]] isomorphic to '''Z''' for {{nowrap|1=''n'' = 2}} or '''Z'''<sub>2</sub> for {{nowrap|''n'' > 2}}.

=== Complex case ===

The general linear group over the field of [[complex number]]s, {{nowrap|GL(''n'', '''C''')}}, is a ''complex'' [[Lie group]] of complex dimension ''n''<sup>2</sup>. As a real Lie group (through realification) it has dimension 2''n''<sup>2</sup>. The set of all real matrices forms a real Lie subgroup. These correspond to the inclusions
:GL(''n'', '''R''') < GL(''n'', '''C''') <  GL(''2n'', '''R'''),
which have real dimensions ''n''<sup>2</sup>, 2''n''<sup>2</sup>, and {{nowrap|1=4''n''<sup>2</sup> = (2''n'')<sup>2</sup>}}. Complex ''n''-dimensional matrices can be characterized as real 2''n''-dimensional matrices that preserve a [[linear complex structure]] &mdash; concretely, that commute with a matrix ''J'' such that {{nowrap|1=''J''<sup>2</sup> = −''I''}}, where ''J'' corresponds to multiplying by the imaginary unit ''i''.

The [[Lie algebra]] corresponding to {{nowrap|GL(''n'', '''C''')}} consists of all {{nowrap|''n''×''n''}} complex matrices with the [[commutator]] serving as the Lie bracket.

Unlike the real case, {{nowrap|GL(''n'', '''C''')}} is [[connected space|connected]]. This follows, in part, since the multiplicative group of complex numbers '''C'''<sup>∗</sup> is connected. The group manifold {{nowrap|GL(''n'', '''C''')}} is not compact; rather its [[maximal compact subgroup]] is the [[unitary group]] U(''n''). As for U(''n''), the group manifold {{nowrap|GL(''n'', '''C''')}} is not [[simply connected]] but has a [[fundamental group]] isomorphic to '''Z'''.

== Over finite fields ==
[[File:Symmetric group 3; Cayley table; GL(2,2).svg|thumb|[[Cayley table]] of {{nowrap|GL(2, 2)}}, which is isomorphic to [[Dihedral group of order 6|S<sub>3</sub>]].]]
If ''F'' is a [[finite field]] with ''q'' elements, then we sometimes write {{nowrap|GL(''n'', ''q'')}} instead of {{nowrap|GL(''n'', ''F'')}}. When ''p'' is prime, {{nowrap|GL(''n'', ''p'')}} is the [[outer automorphism group]]  of the group '''Z'''{{sub|''p''}}{{sup|''n''}}, and also the [[automorphism]] group, because '''Z'''{{sub|''p''}}{{sup|''n''}} is abelian, so the [[inner automorphism group]] is trivial.

The order of {{nowrap|GL(''n'', ''q'')}} is: 
: <math>\prod_{k=0}^{n-1}(q^n-q^k)=(q^n - 1)(q^n - q)(q^n - q^2)\ \cdots\ (q^n - q^{n-1}).</math>

This can be shown by counting the possible columns of the matrix: the first column can be anything but the zero vector; the second column can be anything but the multiples of the first column; and in general, the ''k''th column can be any vector not in the [[linear span]] of the first {{nowrap|''k'' − 1}} columns. In [[q-analog|''q''-analog]] notation, this is <math>[n]_q!(q-1)^n q^{n \choose 2}</math>.

For example, {{nowrap|GL(3, 2)}} has order {{nowrap|1=(8 − 1)(8 − 2)(8 − 4) = 168}}. It is the automorphism group of the [[Fano plane]] and of the group '''Z'''{{sub|2}}{{sup|3}}, and is also known as {{nowrap|[[PSL(2,7)|PSL(2, 7)]]}}.

More generally, one can count points of [[Grassmannian]] over ''F'': in other words the number of subspaces of a given dimension ''k''. This requires only finding the order of the [[stabilizer (group theory)|stabilizer]] subgroup of one such subspace and dividing into the formula just given, by the [[orbit-stabilizer theorem]].

These formulas are connected to the [[Schubert decomposition]] of the Grassmannian, and are [[q-analog|''q''-analogs]] of the [[Betti number]]s of complex Grassmannians. This was one of the clues leading to the [[Weil conjectures]].

Note that in the limit {{nowrap|''q'' ↦ 1}} the order of {{nowrap|GL(''n'', ''q'')}} goes to 0! &ndash; but under the correct procedure (dividing by {{nowrap|(''q'' &minus; 1)<sup>''n''</sup>}}) we see that it is the order of the symmetric group (See Lorscheid's article) &ndash; in the philosophy of the [[field with one element]], one thus interprets the [[symmetric group]] as the general linear group over the field with one element: {{nowrap|''S''<sub>n</sub> ≅ GL(''n'', 1)}}.

=== History ===
The general linear group over a prime field, {{nowrap|GL(''ν'', ''p'')}}, was constructed and its order computed by [[Évariste Galois]] in 1832, in his last letter (to Chevalier) and second (of three) attached manuscripts, which he used in the context of studying the [[Galois group]] of the general equation of order ''p''<sup>''ν''</sup>.<ref name="chevalier-letter">{{cite journal
 | last = Galois
 | first = Évariste
 | year = 1846
 | title = Lettre de Galois à M. Auguste Chevalier
 | journal = [[Journal de Mathématiques Pures et Appliquées]]
 | volume = XI
 | pages = 408&ndash;415
 | url = http://visualiseur.bnf.fr/ark:/12148/cb343487840/date1846
 | accessdate = 2009-02-04
 | postscript =, GL(''ν'',''p'') discussed on p. 410.}}</ref>

== Special linear group ==
{{main article|Special linear group}}

The special linear group, {{nowrap|SL(''n'', ''F'')}}, is the group of all matrices with [[determinant]] 1. They are special in that they lie on a [[Algebraic variety|subvariety]] &ndash; they satisfy a polynomial equation (as the determinant is a polynomial in the entries). Matrices of this type form a group as the determinant of the product of two matrices is the product of the determinants of each matrix. {{nowrap|SL(''n'', ''F'')}} is a [[normal subgroup]] of {{nowrap|GL(''n'', ''F'')}}.

If we write ''F''<sup>×</sup> for the [[multiplicative group]] of ''F'' (excluding 0), then the determinant is a [[group homomorphism]]
:det: GL(''n'', ''F'') → ''F''<sup>×</sup>.
that is surjective and its [[kernel (algebra)|kernel]] is the special linear group. Therefore, by the [[first isomorphism theorem]], {{nowrap|GL(''n'', ''F'')/SL(''n'', ''F'')}} is [[isomorphic]] to ''F''<sup>×</sup>. In fact, {{nowrap|GL(''n'', ''F'')}} can be written as a [[semidirect product]]:
:GL(''n'', ''F'') = SL(''n'', ''F'') ⋊ ''F''<sup>×</sup>

The special linear group is also the [[derived group]] (also known as commutator subgroup) of the GL(''n'', ''F'') (for a field or a [[division ring]] ''F'') provided that <math>n \ne 2</math> or ''k'' is not the [[finite field|field with two elements]].<ref>{{citation|author=Suprunenko|first=D.A.|title=Matrix groups|publisher=American Mathematical Society|year=1976|series=Translations of Mathematical Monographs}}, Theorem II.9.4</ref>

When ''F'' is '''R''' or '''C''', {{nowrap|SL(''n'', ''F'')}} is a [[Lie subgroup]] of {{nowrap|GL(''n'', ''F'')}} of dimension {{nowrap|''n''<sup>2</sup> − 1}}. The [[Lie algebra]] of {{nowrap|SL(''n'', ''F'')}} consists of all {{nowrap|''n''×''n''}} matrices over ''F'' with vanishing [[trace (matrix)|trace]]. The Lie bracket is given by the [[commutator]].

The special linear group {{nowrap|SL(''n'', '''R''')}} can be characterized as the group of ''[[volume]] and [[orientation (mathematics)|orientation]] preserving'' linear transformations of '''R'''<sup>''n''</sup>.

The group {{nowrap|SL(''n'', '''C''')}} is simply connected, while {{nowrap|SL(''n'', '''R''')}} is not. {{nowrap|SL(''n'', '''R''')}} has the same fundamental group as {{nowrap|GL<sup>+</sup>(''n'', '''R''')}}, that is, '''Z''' for {{nowrap|1=''n'' = 2}} and '''Z'''<sub>2</sub> for {{nowrap|''n'' > 2}}.

== Other subgroups ==

=== Diagonal subgroups ===

The set of all invertible [[diagonal matrix|diagonal matrices]] forms a subgroup of {{nowrap|GL(''n'', ''F'')}} isomorphic to (''F''<sup>×</sup>)<sup>''n''</sup>. In fields like '''R''' and '''C''', these correspond to rescaling the space; the so-called dilations and contractions.

A '''scalar matrix''' is a diagonal matrix which is a constant times the [[identity matrix]]. The set of all nonzero scalar matrices forms a subgroup of {{nowrap|GL(''n'', ''F'')}} isomorphic to ''F''<sup>×</sup> . This group is the [[center of a group|center]] of {{nowrap|GL(''n'', ''F'')}}. In particular, it is a normal, abelian subgroup.

The center of {{nowrap|SL(''n'', ''F'')}} is simply the set of all scalar matrices with unit determinant, and is isomorphic to the group of ''n''th [[roots of unity]] in the field ''F''.

=== Classical groups ===

The so-called [[classical group]]s are subgroups of GL(''V'') which preserve some sort of [[bilinear form]] on a vector space ''V''. These include the
* '''[[orthogonal group]]''', O(''V''), which preserves a [[non-degenerate]] [[quadratic form]] on ''V'',
* '''[[symplectic group]]''', Sp(''V''), which preserves a [[Symplectic vector space|symplectic form]] on ''V'' (a non-degenerate [[alternating form]]),
* '''[[unitary group]]''', U(''V''), which, when {{nowrap|1=''F'' = '''C'''}}, preserves a non-degenerate [[hermitian form]] on ''V''.
These groups provide important examples of Lie groups.

== Related groups and monoids ==

=== Projective linear group ===
{{main article|Projective linear group}}
The [[projective linear group]] {{nowrap|PGL(''n'', ''F'')}} and the [[projective special linear group]] {{nowrap|PSL(''n'', ''F'')}} are the [[quotient group|quotients]] of {{nowrap|GL(''n'', ''F'')}} and {{nowrap|SL(''n'', ''F'')}} by their [[Group center|centers]] (which consist of the multiples of the identity matrix therein); they are the induced [[Group action (mathematics)|action]] on the associated [[projective space]].

=== Affine group ===
{{main article|Affine group}}

The [[affine group]] {{nowrap|Aff(''n'', ''F'')}} is an [[group extension|extension]] of {{nowrap|GL(''n'', ''F'')}} by the group of translations in ''F''<sup>''n''</sup>. It can be written as a [[semidirect product]]:
:Aff(''n'', ''F'') = GL(''n'', ''F'') ⋉ ''F''<sup>''n''</sup>
where {{nowrap|GL(''n'', ''F'')}} acts on ''F''<sup>''n''</sup> in the natural manner. The affine group can be viewed as the group of all [[affine transformation]]s of the [[affine space]] underlying the vector space ''F''<sup>''n''</sup>.

One has analogous constructions for other subgroups of the general linear group: for instance, the [[special affine group]] is the subgroup defined by the semidirect product, {{nowrap|SL(''n'', ''F'') ⋉ ''F''<sup>''n''</sup>}}, and the [[Poincaré group]] is the affine group associated to the [[Lorentz group]], {{nowrap|O(1, 3, ''F'') ⋉ ''F''<sup>''n''</sup>}}.

=== General semilinear group ===
{{main article|General semilinear group}}
The [[general semilinear group]] {{nowrap|ΓL(''n'', ''F'')}} is the group of all invertible [[semilinear transformation]]s, and contains GL. A semilinear transformation is a transformation which is linear "up to a twist", meaning "up to a [[field automorphism]] under scalar multiplication". It can be written as a semidirect product:
:ΓL(''n'', ''F'') = Gal(''F'') ⋉ GL(''n'', ''F'')
where Gal(''F'') is the [[Galois group]] of ''F'' (over its [[prime field]]), which acts on {{nowrap|GL(''n'', ''F'')}} by the Galois action on the entries.

The main interest of {{nowrap|ΓL(''n'', ''F'')}} is that the associated [[projective semilinear group]] {{nowrap|PΓL(''n'', ''F'')}} (which contains {{nowrap|PGL(''n'', ''F''))}} is the [[collineation group]] of [[projective space]], for {{nowrap|''n'' > 2}}, and thus semilinear maps are of interest in [[projective geometry]].

=== Full linear monoid ===
{{expand section|basic properties|date=April 2015}}
If one removes the restriction of the determinant being non-zero, the resulting algebraic structure is a [[monoid]], usually called the '''full linear monoid''',<ref name="Okniński1998">{{cite book|author=Jan Okniński|title=Semigroups of Matrices|year=1998|publisher=World Scientific|isbn=978-981-02-3445-4|at=Chapter 2: Full linear monoid}}</ref><ref name="Meakin">{{cite book|editor=C. M. Campbell|title=Groups St Andrews 2005|year=2007|publisher=Cambridge University Press|isbn=978-0-521-69470-4|page=471|chapter=Groups and Semigroups: Connections and contrast|author=Meakin}}</ref><ref name="RhodesSteinberg2009">{{cite book|author1=John Rhodes|author2=Benjamin Steinberg|title=The q-theory of Finite Semigroups|year=2009|publisher=Springer Science & Business Media|isbn=978-0-387-09781-7|page=306}}</ref> but occasionally also ''full linear semigroup'',<ref name="JespersOkniski2007">{{cite book|author1=Eric Jespers|author2=Jan Okniski|title=Noetherian Semigroup Algebras|year=2007|publisher=Springer Science & Business Media|isbn=978-1-4020-5810-3|at=2.3: Full linear semigroup}}</ref> ''general linear monoid''<ref name="Geck2013">{{cite book|author=Meinolf Geck|title=An Introduction to Algebraic Geometry and Algebraic Groups|year=2013|publisher=Oxford University Press|isbn=978-0-19-967616-3|page=132}}</ref><ref name="CanLi2014">{{cite book|author1=Mahir Bilen Can|author2=Zhenheng Li|author3=Benjamin Steinberg|author4=Qiang Wang|title=Algebraic Monoids, Group Embeddings, and Algebraic Combinatorics|year=2014|publisher=Springer|isbn=978-1-4939-0938-4|page=142}}</ref> etc. It is actually a [[regular semigroup]].<ref name="Meakin"/>

== Infinite general linear group ==
The '''infinite general linear group''' or '''[[direct limit of groups|stable]] general linear group''' is the [[direct limit]] of the inclusions {{nowrap|GL(''n'', ''F'') → GL(''n'' + 1, ''F'')}} as the upper left [[block matrix]]. It is denoted by either GL(''F'') or {{nowrap|GL(∞, ''F'')}}, and can also be interpreted as invertible infinite matrices which differ from the identity matrix in only finitely many places.<ref name=Mil25>{{cite book | last1=Milnor | first1=John Willard | author1-link= John Milnor | title=Introduction to algebraic K-theory | publisher=[[Princeton University Press]] | location=Princeton, NJ | mr=0349811 | year=1971 | zbl=0237.18005 | series=Annals of Mathematics Studies | volume=72 | page=25 }}</ref>

It is used in [[algebraic K-theory]] to define [[Algebraic K-theory#K1|K<sub>1</sub>]], and over the reals has a well-understood topology, thanks to [[Bott periodicity]].

It should not be confused with the space of (bounded) invertible operators on a [[Hilbert space]], which is a larger group, and topologically much simpler, namely contractible &ndash; see [[Kuiper's theorem]].

== See also ==
* [[List of finite simple groups]]
* [[SL2(R)|SL<sub>2</sub>('''R''')]]
* [[Representation theory of SL2(R)|Representation theory of SL<sub>2</sub>('''R''')]]

== Notes ==
{{reflist|2}}

== External links ==
*{{springer|title=General linear group|id=p/g043680}}
* [http://demonstrations.wolfram.com/GL2PAndGL33ActingOnPoints/ "GL(2, ''p'') and GL(3, 3) Acting on Points"] by [[Ed Pegg, Jr.]], [[Wolfram Demonstrations Project]], 2007.

[[Category:Abstract algebra]]
[[Category:Linear algebra]]
[[Category:Lie groups]]
[[Category:Linear algebraic groups]]