Matrix groups with real entries

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.

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