Editing General linear group (section)

== As a Lie group ==

=== Real case ===

The general linear group <math>\operatorname{GL}(n,\R)</math> over the field of [[real number]]s is a real [[Lie group]] of dimension <math>n^2</math>. To see this, note that the set of all <math>n\times n</math> real matrices, <math>M_n(\R)</math>, forms a [[real vector space]] of dimension <math>n^2</math>. The subset <math>\operatorname{GL}(n,\R)</math> consists of those matrices whose [[determinant]] is non-zero. The determinant is a [[polynomial]] map, and hence <math>\operatorname{GL}(n,\R)</math> is an [[algebraic variety|open affine subvariety]] of <math>M_n(\R)</math> (a [[non-empty]] [[open subset]] of <math>M_n(\R)</math> 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 <math>\operatorname{GL}(n,\R)</math>, denoted <math>\mathfrak{gl}_n,</math> consists of all <math>n\times n</math> real matrices with the [[commutator]] serving as the Lie bracket.

As a manifold, <math>\operatorname{GL}(n,\R)</math> 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 <math>\operatorname{GL}^+(n,\R)</math>, consists of the real <math>n\times n</math> matrices with positive determinant. This is also a Lie group of dimension <math>n^2</math>; it has the same Lie algebra as <math>\operatorname{GL}(n,\R)</math>.

The [[polar decomposition]], which is unique for invertible matrices, shows that there is a homeomorphism between <math>\operatorname{GL}(n,\R)</math> and the Cartesian product of <math>\operatorname{O}(n)</math> with the set of positive-definite symmetric matrices. Similarly, it shows that there is a homeomorphism between <math>\operatorname{GL}^+(n,\R)</math> and the Cartesian product of <math>\operatorname{SO}(n)</math> with the set of positive-definite symmetric matrices. Because the latter is contractible, the [[fundamental group]] of <math>\operatorname{GL}^+(n,\R)</math> is isomorphic to that of <math>\operatorname{SO}(n)</math>.

The homeomorphism also shows that the group <math>\operatorname{GL}(n,\R)</math> is [[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 <math>\operatorname{GL}(n,\R)</math> is the [[orthogonal group]] <math>\operatorname{O}(n)</math>, while "the" maximal compact subgroup of <math>\operatorname{GL}^+(n,\R)</math> is the [[special orthogonal group]] <math>\operatorname{SO}(n)</math>. As for <math>\operatorname{SO}(n)</math>, the group <math>\operatorname{GL}^+(n,\R)</math> is not [[simply connected]] (except when <math>n=1</math>, but rather has a [[fundamental group]] isomorphic to <math>\Z</math> for <math>n=2</math> or <math>\Z_2</math> for <math>n>2</math>.

=== Complex case ===

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

The [[Lie algebra]] corresponding to <math>\operatorname{GL}(n,\C)</math> consists of all <math>n\times n</math> complex matrices with the [[commutator]] serving as the Lie bracket.

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