Name

Dim

Type

Rank

Matrices

Conditions

Examples

sl(n)

 $n^2-1$

A

$n -1$

$\left[A\right]$

$\mathrm{tr}\left(A\right) \= 0$

Example 1.

$\mathrm{su}\left(n\right)$

$n^{2 }-1$

A

$n-1$

$\left[\begin{array}{r}Z\end{array}\right] \= \left[\begin{array}{r}{A}_1 \+I A_2\end{array}\right]$

$Z \+ Z^{†} \= 0 \, A_1 \+ A_1^t \= 0\, A_2 - A_2^t \= 0.$

Example 2

$\mathrm{su}\left(p\,q\right)$

version 1

$n^2 -1$

A

 $n-1$

$\left[\begin{array}{rrr}{Z}_1& Z_2& Z_3\\ Z_4& -Z_1^{†}& Z_5\\ -Z_5& -Z_3& Z_6\end{array}\right] \= \left[\begin{array}{rrr}{A}_1\+{\mathrm{IA}}_2& B_1\+ {\mathrm{IB}}_2& C_1 \+ {\mathrm{IC}}_2\\ {\mathrm{D}}_1 \+ I {\mathrm{D}}_2^t& -A_1\+{\mathrm{IA}}_{2^{}}^t & E_1 \+{\mathrm{IE}}_2\\ E_1 \+ {\mathrm{IE}}_{2^{}}^t& -C_1 \+C_{2^{}}^t & F_1\+{\mathrm{IF}}_{2^{}}^t\end{array}\right]$

$Z_1\, Z_2 \, Z_4 q \times q \; Z_3 \, Z_5 q\times \left(p-q\right)\; Z_6 \left(p-q\right)\times \left(p-q\right)$

$Z_1$, $Z_3 \,Z_5$ arbitrary, $Z_2\+Z_2^{†}\=0\,$$Z_4 \+{Z_4^{†}}^{}\=0\,$$Z_6 \+{Z_6^{†}}^{}\=0\, \mathrm{tr}\left(Z_6\right)\= 0$

$B_1\, {\mathrm{D}}_1\, F_1$skew-symmetric

$B_2\, {\mathrm{D}}_2\, F_2$symmetric

Example 3

$\mathrm{su}\left(p\, q\right)$

version 2

$n^{2 }-1$

$n \= p\+q$

A

$n-1$

$\left[\begin{array}{rr}{Z}_1& Z_2\\ Z_2^{†}& Z_3\end{array}\right]\= \left[\begin{array}{rr}{A}_1\+ I A_2& B_1 \+ I B_2\\ B_1^t - I B_2^t& C_1\+{\mathrm{IC}}_2 \end{array}\right]$

$Z_1 p\times p$$\,$$Z_2 q\times q$$\,$$Z_3 p\times q$

$Z_1^{} \+ Z_1^{†} \=0\, Z_3 \+ Z_3^{†} \=0\, \mathrm{tr}\left(Z_1\right) \+ \mathrm{tr}\left(Z_3\right) \=0\, Z_2$arbitrary

$A_1 \+ A_1^t \= 0\, A_2 - A_2^t \= 0\,$ $C_1 \+ C_1^t \= 0\, C_2 - C_2^t \=0\,$$\mathrm{tr}\left(A_1\right) \+ \mathrm{tr}\left(C_1\right)\= 0$

Example 3

${\mathrm{su}}^{\*}\left(n\right)$

$n \mathrm{even}$

$n^{2 }- 1$

A

$n-1$

$\left[\begin{array}{rr}{Z}_1& Z_2\\ -\overline{Z_2}& {\bar{Z}}_1\end{array}\right] \= \left[\begin{array}{rr}{A}_1 \+ I A_2& B_1 \+ I B_2\\ -B_1 \+ I B_2& A_{1 }- I A_2\end{array}\right]$

$\mathrm{tr}\left(Z_1\right) \+ \mathrm{tr}\left(\overline{Z_1}\right) \= 0\, \mathrm{tr}\left(A_1\right) \=0.$

Example 4

$\mathrm{so}\left(p\,q\right)$

$p \+q \= n\,$$n\=2 m \+1$

version 1

$\frac{1}{2} n\left(n-1\right)$

B

$m$

$\left[\begin{array}{rrr}A& B& C\\ \mathrm{D}& -A^t& E\\ -E^{}& -C^t& F\end{array}\right]$

$A\, B\, \mathrm{D} q\times q\; C\,E\, q\times \left(p-q\right)\; F \left(p-q\right)\times \left(p-q\right)$

$A\, C\, E$arbitrary

$B\+ B^t \=0\, \mathrm{D} \+ {\mathrm{D}}^t\=0$, $F \+ F^t\=0$

Example 5

$\mathrm{so}\left(p\,q\right)$

$p \+q \= n\,$$n\=2 m \+1$

version 2

$\frac{1}{2} n\left(n-1\right)$

B

$m$

$\left[\begin{array}{rr}A& B\\ -B^t& C\end{array}\right]$

A $p\times p\; B p\times q\; C q\times q$

$A \+ A^t\=0\, C \+ C^{t }\=0\, B$arbitrary

Example 5

$\mathrm{sp}\left(n\, \mathrm{\ℝ}\right)$

$n \= 2 m$

$n\left(n \+1\right)$

C

$m$

$\left[\begin{array}{rr}A& B\\ C& -A^t\end{array}\right]$

$A\,B\,C m\times m$

$B - B^t \=0\, B - B^t \=0$

Example 6

$\mathrm{sp}\left(p\, q\right)$

$2 p \+2 q \= n$

$n\left(n \+1\right)$

C

$m$

$\left[\begin{array}{rrrr}{Z}_1& Z_2& Z_3& Z_4\\ Z_2^{†}& Z_5& Z_{4^{}}^t& Z_6\\ -\overline{Z_3}& \overline{Z_4}& \overline{Z_1}& -\overline{Z_2}\\ Z_4^{†}& -\stackrel{}{Z_6}& -Z_{2^{}}^t& \overline{Z_5}\end{array}\right]\= \left[\begin{array}{rrrr}{A}_1 \+{\mathrm{IA}}_2& B_1 \+{\mathrm{IB}}_2& C_1 \+{\mathrm{IC}}_2& {\mathrm{D}}_1 \+{\mathrm{ID}}_2\\ B_1^t -I B_2^t& E_1 \+{\mathrm{IE}}_2& {\mathrm{D}}_1^t \+{\mathrm{ID}}_2^t& F_1 \+{\mathrm{IF}}_2\\ -C_1 \+{\mathrm{IC}}_2& {\mathrm{D}}_1 -{\mathrm{ID}}_2& A_1 - {\mathrm{IA}}_2& -B_1 \+{\mathrm{IB}}_2\\ {\mathrm{D}}_1 \+{\mathrm{ID}}_2& -F_1 \+{\mathrm{IF}}_2& -B_{{}^{}1}^t \+{\mathrm{IB}}_2^t& E_1 - {\mathrm{IE}}_2\end{array}\right]$

$Z_1\, Z_3 p\times p\, Z_2\, Z_{4 }p\times q\, Z_5\, Z_6 q\times q$

$Z_1 \+Z_1^{†} \= 0\, Z_5 \+Z_5^{†} \=0\, Z_2\, Z_4$arbitrary

$Z_{3 }- Z_{3^{}}^t \=0\, Z_6- Z_6^t$ = 0,

$A_1 \+A_1^t \=0\, A_2 - A_2^t \=0\, E_1 \+E_1^t \=0 \, E_2 - E_2^t \=0\,$

$C_1 - C_1^t \=0$, $C_1 \+ C_1^t \=0$, $F_1- F_1^t \=0$, $F_1 \+ F_1^t \=0$ 

Example 7

$\mathrm{sp}\left(n\right)$

$n \= 2 m$

$n\left(n\+1\right)$

C

$m$

$\left[\begin{array}{rr}{Z}_1& Z_2\\ -{\bar{Z}}_2& \overline{Z_1}\end{array}\right]\= \left[\begin{array}{rr}{A}_1 \+ {\mathrm{IA}}_2& B_1\+ {\mathrm{IB}}_2\\ -B_1\+ {\mathrm{IB}}_2& A_1 - {\mathrm{IA}}_2\end{array}\right]$

$Z_1\, Z_2 m\times m$

$Z_1 \+ Z_1^{†} \= 0\, Z_{2 }- Z_2^t \= 0\,$

$A_1 \+A_1^t\=0\, A_1- A_1^t\=0\, B_1 -B_1^t \=0\, B_2- B_2^t \=0\,$

Example 7

$\mathrm{so}\left(p\, q\right)$

$p \+q \= n$

$n\=2 m$

version 1

$\frac{1}{2} n\left(n-1\right)$

D

$m$

$\left[\begin{array}{rrr}A& B& C\\ \mathrm{D}& -A^t& E\\ -E^{}& -C^t& F\end{array}\right]$

$A\, B\, \mathrm{D} q\times q\; C\,E\, q\times \left(p-q\right)\; F \left(p-q\right)\times \left(p-q\right)$

$A\, C\, E$arbitrary

$B\+ B^{t }\=0\, \mathrm{D} \+ {\mathrm{D}}^t\=0$, $F \+ F^t \=0$

Example 8

$\mathrm{so}\left(p\, q\right)$

$p \+q \= n$

$n\=2 m \+1$

version 2

$\frac{1}{2} n\left(n-1\right)$

D

$m$

$\left[\begin{array}{rr}A& B\\ -B^t& C\end{array}\right]$

A $p\times p\; B p\times q\; C q\times q$

$A \+ A^t\=0\, C \+ C^t\=0\, B$arbitrary

Example 9

${\mathrm{so}}^{\*}\left(n\right)$

$n \= 2 m$

$\frac{1}{2} n\left(n-1\right)$

D

$m$

$\left[\begin{array}{rr}{Z}_1& Z_2\\ -\overline{Z_2}& {\bar{Z}}_1\end{array}\right] \= \left[\begin{array}{rr}{A}_1\+{\mathrm{IA}}_2& B_1\+I B_2\\ -B^{t }\+I B_2^t& A_{1 }- {\mathrm{IA}}_2\end{array}\right]$

$Z_1\, Z_2 m\times m$

$Z_1 \+Z_1^t \=0\, Z_{2 }- Z_2^{†} \= 0$

$A_1 \+A_1^t \=0$, $A_2\+A_2^t \=0\, B_1-B_1^t \=0\, B_2 \+ B_2^t \=0$

Example 10 

Name

Dim

Matrices

Conditions

Examples

$\mathrm{gl}\left(n\,\mathrm{\ℝ}\right)$

$n^2$

$\left[\begin{array}{r}A\end{array}\right]$

Z, $A_1 \, A_2 n\times n$

Example 11

$\mathrm{gl}\left(n\, \mathrm{\ℂ}\right)$

$2 n^2$

$\left[\begin{array}{r}Z\end{array}\right] \= \left[\begin{array}{r}{A}_1 \+ I A_2\end{array}\right]$

Z, $A_1 \, A_2 n\times n$

Example 11

$\mathrm{sl}\left(n\,\mathrm{\ℂ}\right)$

$2\left(n^2 -1\right)$

$\left[\begin{array}{r}Z\end{array}\right] \= \left[\begin{array}{r}{A}_1 \+ I A_2\end{array}\right]$

Z, $A_1 \, A_2 n\times n$

Z, $A_1 \, A_2$trace-free

Example 12

$u\left(p\, q\right)$

$n^2$

$\left[\begin{array}{rr}{Z}_1& Z_2\\ Z_2^{†}& Z_3\end{array}\right]\= \left[\begin{array}{rr}{A}_1\+ I A_2& B_1 \+ I B_2\\ B_1^t - I B_2^t& C_1\+{\mathrm{IC}}_2 \end{array}\right]$

$Z_1 p\times p\,$$Z_2 q\times q\,$$Z_3 p\times q$

$Z_1^{} \+ Z_1^{†} \=0\, Z_3\+ Z_3^{†} \=0\, Z_2$arbitrary

$A_1 \+ A_1^t \= 0\, A_2 - A_2^t \= 0\,$ $C_1 \+ C_1^t \= 0\, C_2 - C_2^t \=0\,$

Example 13

$\mathrm{so}\left(n\,\mathrm{\ℂ}\right)$

$n\left(n-1\right)$

$\left[\begin{array}{r}Z\end{array}\right] \= \[A_1\+{\mathrm{IA}}_2\]$

Z, $A_1 \, A_2 n\times n$

Z, $A_1\, A_{2 }$skew-symmetric

Example 14

$\mathrm{sp}\left(n\, \mathrm{\ℂ}\right)$

$2 n\left(n \+1\right)$

$\left[\begin{array}{rr}{Z}_1& Z_2\\ Z_3& -Z_1^t\end{array}\right] \=$$\left[\begin{array}{rr}{A}_1 \+ {\mathrm{IA}}_2& B_1\+{\mathrm{IB}}_2\\ C_1\+{\mathrm{IC}}_2& -A^t - {\mathrm{IA}}_2^t\end{array}\right]$

$Z_1\, Z_2\,Z_3\, A_1\, A_2\, B_1\, B_2\, C_1\, C_2 n\times n$

$Z_2 \+ Z_{2^{}}^t \=0\, Z_3 \+ Z_3^t \=0\,$

$B_1\+B_1^t \=0\, B_2\+B_2^t \=0\, C_1 \+C_1^t \=0\, C_2\+C_2^t \=0$

 Example 15

$\mathrm{sol}\left(n\right)$

$\frac{1}{2}n\left(n\+1\right)$

$\left[\begin{array}{r}A\end{array}\right]$

$A n\times n$

upper triangular

Example 16

$\mathrm{nil}\left(n\right)$

$\frac{1}{2}n\left(n-1\right)$

$\left[\begin{array}{r}A\end{array}\right]$

$A n\times n$

strictly upper triangular

Example 17