Matrix Groups

$$\begin{align*} \operatorname{GL}_n({\mathbf{R}}) = \left\{{ A {~\mathrel{\Big\vert}~}A = \overline{A} }\right\} .\end{align*}$$

$$\begin{align*} {\operatorname{SL}}_n({\mathbf{C}}) \coloneqq\left\{{ A {~\mathrel{\Big\vert}~}\operatorname{det}A = 1 }\right\} .\end{align*}$$

$$\begin{align*} O_n({\mathbf{C}}) \coloneqq\left\{{ A {~\mathrel{\Big\vert}~}A^tA = A A^t = I}\right\} .\end{align*}$$

Dimension: $n(n-1) / 2$.

$$\begin{align*} {\operatorname{SO}}_n({\mathbf{R}}) = \left\{{ A {~\mathrel{\Big\vert}~}AA^t = I}\right\} = \ker(\operatorname{GL}_n({\mathbf{R}}) \to k^{\times}) .\end{align*}$$

$$\begin{align*} U_n({\mathbf{C}}) \coloneqq\left\{{ A {~\mathrel{\Big\vert}~}A^\dagger A = AA^\dagger = 1 }\right\} .\end{align*}$$

$$\begin{align*} {\operatorname{SU}}_n({\mathbf{C}}) \coloneqq\left\{{ A \in U_n({\mathbf{C}}) {~\mathrel{\Big\vert}~}\operatorname{det}A = 1 }\right\} .\end{align*}$$

$$\begin{align*} \mathrm{Sp}_{2n}({\mathbf{C}}) \coloneqq\left\{{ A \in \operatorname{GL}_{2n}({\mathbf{C}}) {~\mathrel{\Big\vert}~}A^tJA = J }\right\} && J \coloneqq \begin{bmatrix} 0 & 1_n \\ 1_n & 0 \end{bmatrix} .\end{align*}$$