Editing General linear group (section)

=== 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>.