# Matrix groups

The most common Lie groups are the matrix groups (AKA linear groups), which are Lie subgroups of the group of real or complex $${n\times n}$$ invertible matrices, denoted $${GL(n,\mathbb{R})}$$ and $${GL(n,\mathbb{C})}$$. We can also consider the linear groups, which are Lie subgroups of $${GL(V)}$$, the group of invertible linear transformations on a real or complex vector space $${V}$$. One can then choose a basis of $${V}$$ to get a (non-canonical) isomorphism to the matrix group, e.g. from $${GL(\mathbb{R}^{n})}$$ to $${GL(n,\mathbb{R})}$$.

 Δ The distinction between the abstract linear groups and the basis-dependent matrix groups is not always made, and the notation is used interchangeably. Alternative notation includes $${GL_{n}(\mathbb{R})}$$, and the field and/or dimension is often omitted, yielding notation such as $${GL_{n}}$$, $${GL(n)}$$, $${GL(\mathbb{R})}$$, or $${GL}$$.

These groups can be seen to be Lie groups by taking global coordinates to be the real matrix entries or the real components of the complex entries. Thus $${GL(n,\mathbb{R})}$$ is a manifold of dimension $${n^{2}}$$, and $${GL(n,\mathbb{C})}$$ has manifold dimension $${2n^{2}}$$. Any subgroup of $${GL}$$ that is also a submanifold is then automatically a Lie subgroup.

We can also consider Lie groups defined by invertible matrices with entries in $${\mathbb{H}}$$ or $${\mathbb{O}}$$, since even though they cannot be viewed as linear transformations on a vector space, they still form a group and are manifolds with respect to the real components of their entries.

 Δ Some matrix groups can also be viewed as a complex Lie group, a group that is also a complex manifold. For example, $${GL(n,\mathbb{C})}$$ can be viewed as an $${n^{2}}$$-dimensional complex Lie group instead of as a real Lie group of dimension $${2n^{2}}$$. It is important to distinguish between a complex Lie group and a real Lie group defined by matrices with complex entries.