Complex matrices:
Number field $\mathbb{F}$	Real $\mathbb{R}$ or complex $\mathbb{C}$
$M(d,\mathbb{F})$	Space of square $d\times d$ matrices in $\mathbb{F}$
$\mathrm{Sym}(d,\mathbb{R})$	Space of real symmetric matrices
0	matrix with all coefficients equal to zero (disk origin)
Fröbenius norm	${\parallel M\parallel }_F=\sqrt{{\sum }_{i,j}{\left|M_{i,j}\right|}^2}$
Operator norm	${\parallel M\parallel }_O={\sigma }_{\max }\left(M\right)={\max }_i\left\{\right|{\lambda }_i\left(M\right)\left|\right\}$
Domains:
Cone of SPD matrices	$\mathrm{PD}(d,\mathbb{R})=\{P\succ 0:P\in \mathrm{Sym}(d,\mathbb{R}\left)\right\}$
Siegel–Poincaré upper plane	$\mathbb{SH}\left(d\right)=\left\{Z=X+iY:X\in \mathrm{Sym}(d,\mathbb{R}),Y\in \mathrm{PD}(d,\mathbb{R})\right\}$
Siegel–Poincaré disk	$\mathbb{SD}\left(d\right)=\left\{W\in \mathrm{Sym}(d,\mathbb{C}):I-\overline{W}W\succ 0\right\}$
Distances:
Siegel distance	${\rho }_U(Z_1,Z_2)=\sqrt{{\sum }_{i=1}^d{log}^2\left(\frac{1+\sqrt{r_i}}{1-\sqrt{r_i}}\right)}$
	$r_i={\lambda }_i\left(R(Z_1,Z_2)\right)$
	$R(Z_1,Z_2):=(Z_1-Z_2){(Z_1-{\overline{Z}}_2)}^{-1}({\overline{Z}}_1-{\overline{Z}}_2){({\overline{Z}}_1-{\overline{Z}}_2)}^{-1}$
Upper plane metric	$\mathrm{d}{s}_U\left(Z\right)=2\mathrm{tr}\left(Y^{-1}\mathrm{d}Z{Y}^{-1}\mathrm{d}\overline{Z}\right)$
PD distance	${\rho }_{\mathrm{PD}}(P_1,P_2)={\parallel \mathrm{Log}\left(P_1^{-1}{P}_2\right)\parallel }_F=\sqrt{{\sum }_{i=1}^d{log}^2\left({\lambda }_i\left(P_1^{-1}{P}_2\right)\right)}$
PD metric	$\mathrm{d}{s}_{\mathrm{PD}}\left(P\right)=\mathrm{tr}\left({\left(P^{-1}\mathrm{d}P\right)}^2\right)$
Kobayashi distance	${\rho }_D(W_1,W_2)=log\left(\frac{1+\parallel {\Phi }_{W_1}\left(W_2\right){\parallel }_O}{1-\parallel {\Phi }_{W_1}\left(W_2\right){\parallel }_O}\right)$
Translation in the disk	${\Phi }_{W_1}\left(W_2\right)={(I-W_1{\overline{W}}_1)}^{-\frac{1}{2}}(W_2-W_1){(I-{\overline{W}}_1{W}_2)}^{-1}{(I-{\overline{W}}_1{W}_1)}^{\frac{1}{2}}$
Disk distance to origin	${\rho }_D(0,W)=log\left(\frac{1+{\parallel W\parallel }_O}{1-{\parallel W\parallel }_O}\right)$
Siegel–Klein distance	${\rho }_K(K_1,K_2)=\left\{\begin{array}{cc}\frac{1}{2}log\left|\frac{{\alpha }_{+}(1-{\alpha }_{-})}{{\alpha }_{-}({\alpha }_{+}-1)}\right|,\hfill & K_1\ne K_2,\hfill \\ 0\hfill & K_1=K_2\hfill \end{array}\right.$
	$\parallel (1-{\alpha }_{-})K_1+{\alpha }_{-}{K}_2{\parallel }_O=1$ (${\alpha }_{-}<0$), $\parallel (1-{\alpha }_{+})K_1+{\alpha }_{+}{K}_2{\parallel }_O=1$ (${\alpha }_{+}>1$)
Seigel-Klein distance to 0	${\rho }_K(0,K)=\frac{1}{2}log\left(\frac{1+{\parallel K\parallel }_O}{1-{\parallel K\parallel }_O}\right)$
Symplectic maps and groups:
Symplectic map	${\varphi }_S\left(Z\right)=(AZ+B){(CZ+D)}^{-1}$ with $S\in \mathrm{Sp}(d,\mathbb{R})$ (upper plane)
	${\varphi }_S\left(W\right)$ with $S\in \mathrm{Sp}(d,\mathbb{C})$ (disk)
Symplectic group	$\mathrm{Sp}(d,\mathbb{F})=\left\{\left[\begin{array}{cc}A& B\\ C& D\end{array}\right],A{B}^{\top }=B{A}^{\top },C{D}^{\top }=D{C}^{\top },A{D}^{\top }-B{C}^{\top }=I\right\}$
	$A,B,C,D\in M(d,\mathbb{F})$
group composition law	matrix multiplication
group inverse law	$S^{(-1)}=:\left[\begin{array}{cc}{D}^{\top }& -B^{\top }\\ -C^{\top }& A^{\top }\end{array}\right]$
Translation in $\mathbb{H}\left(d\right)$ of $Z=A+iB$ to $iI$	$T_U\left(Z\right)=\left[\begin{array}{cc}{\left(B^{\frac{1}{2}}\right)}^{\top }& 0\\ -{\left(A{B}^{-\frac{1}{2}}\right)}^{\top }& {\left(B^{-\frac{1}{2}}\right)}^{\top }\end{array}\right]$
symplectic orthogonal matrices	$\mathrm{SpO}(2d,\mathbb{R})=\left\{\left[\begin{array}{cc}A& B\\ -B& A\end{array}\right]:A^{\top }A+B^{\top }B=I,A^{\top }B\in \mathrm{Sym}(d,\mathbb{R})\right\}$
(rotations in $\mathbb{SH}\left(d\right)$)
Translation to 0 in $\mathbb{SD}\left(d\right)$	${\Phi }_{W_1}\left(W_2\right)={(I-W_1{\overline{W}}_1)}^{-\frac{1}{2}}(W_2-W_1){(I-{\overline{W}}_1{W}_2)}^{-1}{(I-{\overline{W}}_1{W}_1)}^{\frac{1}{2}}$
${\mathrm{Isom}}^{+}\left(\mathbb{S}\right)$	Isometric orientation-preserving group of generic space $\mathbb{S}$
$\mathrm{Moeb}\left(d\right)$	group of Möbius transformations