Set of invertible matrices with real entries

[LagrangeEuler]

##GL(n,\mathbb{R})## is set of invertible matrices with real entries. We know that
[tex]SO(n,\mathbb{R}) \subset O(n,\mathbb{R}) \subset GL(n,\mathbb{R})[/tex]
is there any specific subgroups of ##GL(n,\mathbb{R})## that is highly important.

 

[jgens]

You named most of the big ones. I would definitely add SL(n,R) to that list. Some of the subgroups consisting of diagonal matrices and say upper triangular matrices are important too.

 

[George Jones]

Other important subgroups are the pseudo-orthogonal groups O(p,g) and SO(p,q) (e.g., the Lorentz group) and the symplectic groups.

 

[jgens]

There is an embedding GL(n,C) < GL(2n,R) so all the important subgroups of GL(n,C) like the unitary groups can be regarded as subgroups of GL(2n,R) as well.

 

[blue_raver22]

Yes, there are several important subgroups of ##GL(n,\mathbb{R})##. One example is the special linear group, ##SL(n,\mathbb{R})##, which consists of matrices with determinant equal to 1. This group is important in many areas of mathematics, including geometry, number theory, and representation theory. Another important subgroup is the orthogonal group, ##O(n,\mathbb{R})##, which consists of matrices that preserve the dot product on ##\mathbb{R}^n##. This group has applications in geometry, physics, and computer graphics. Additionally, the general linear group itself, ##GL(n,\mathbb{R})##, is of fundamental importance in linear algebra and its applications.