# 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 $${\mathbb{R}^{n}}$$.

NameGeometryMatrixLie 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.