2.1 Matrix Groups

A Lie group $\mathrm{G}$ is simultaneously a smooth differentiable manifold and a group. The Lie groups treated in this document are all real matrix groups: group elements are represented as matrices in $\mathbb{R}^{n\times n}$. The groups' multiplication and inversion operations are identically matrix multiplication and inversion. Because each group is represented by a specific subclass of non-singular $n\times n$ matrices, there are fewer than $n^{2}$ degrees of freedom.