Here we summarize some of the common matrix groups with real entries, with a focus on their geometrical properties as linear transformations on \({\mathbb{R}^{n}}\).
| Name | Geometry | Matrix | Lie algebra |
|---|---|---|---|
| \({GL\left(n,\mathbb{R}\right)}\) General linear group | Arbitrary change of basis in \({\mathbb{R}^{n}}\) | \({n\times n}\) matrices with \({\textrm{det}\left(A\right)\neq0}\) | All \({n\times n}\) matrices |
| \({GL\left(n,\mathbb{R}\right)^{e}}\) | Preserves orientation | \({\textrm{det}\left(A\right)>0}\) | \({gl\left(n,\mathbb{R}\right)}\) |
| \({SL\left(n,\mathbb{R}\right)}\) Special linear group | Preserves orientation and volume | \({\textrm{det}\left(A\right)=1}\) | \({\textrm{tr}\left(A\right)=0}\) |
| \({O\left(n\right)}\) Orthogonal group | Preserves the Euclidean inner product: rotations and reflections | \({A^{\textrm{T}}A=I}\) \({\Rightarrow\textrm{det}\left(A\right)=\pm1}\) | \({A^{\textrm{T}}=-A}\) |
| \({SO\left(n\right)}\) Special orthogonal group | Proper rotations (preserves orientation) | \({A^{\textrm{T}}A=I,}\) \({\textrm{det}\left(A\right)=1}\) | \({o\left(n,\mathbb{R}\right)}\) |
| \({O\left(r,s\right)}\) Pseudo-orthogonal group | Preserves the pseudo-Euclidean inner product | \({A^{\textrm{T}}\eta A=\eta}\) \({\Rightarrow\textrm{det}\left(A\right)=\pm1}\) | Matrices \({\eta A}\) for \({A}\) anti-symmetric |
| \({SO\left(r,s\right)}\) Special pseudo-orthogonal group | As above, but preserves orientation | \({A^{\textrm{T}}\eta A=\eta,}\) \({\textrm{det}\left(A\right)=1}\) | \({o\left(r,s\right)}\) |
| \({Sp\left(2n,\mathbb{R}\right)}\) Real symplectic group | Preserves the symplectic form | \({A^{\textrm{T}}JA=J}\) \({\Rightarrow\textrm{det}\left(A\right)=1}\) | \({JA+A^{\textrm{T}}J=0}\) |
Notes: Just as \({GL(n,\mathbb{R})}\) is often written \({GL_{n}}\), similar notation is sometimes used for other groups. The notation does not distinguish between abstract and matrix groups; we will attempt to note the distinction when relevant. \({GL\left(n,\mathbb{R}\right)^{e}}\) is often written \({GL_{n}^{+}}\) or similar. An immediate result from their definitions is \({O\left(r,s\right)\cong O\left(s,r\right)}\) and \({SO\left(r,s\right)\cong SO\left(s,r\right)}\). The notation \({Sp\left(2n,\mathbb{R}\right)}\) reflects the fact that \({J}\) only exists for even-dimensional matrices; however, sometimes it is denoted \({Sp\left(n,\mathbb{R}\right)}\), where the group still consists of \({2n\times2n}\) matrices. We will always use notation consistent with the size of the defining matrices.