# Set of invertible matrices with real entries

1. Dec 4, 2013

### LagrangeEuler

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

Last edited: Dec 4, 2013
2. Dec 4, 2013

### 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.

3. Dec 4, 2013

### George Jones

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

4. Dec 4, 2013

### 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.