# 2 \(\times\) 2 Matrices: Manifolds, Realizations, Applications
## 2 \(\times\) 2 Matrices Covering 3 \(\times\) 3 Matrices
### Pseudo-Unitary Cover Pseudo-Orthogonal Matrices  
Closely related to the cover afforded by \(\mathsf{SU}(2)\) over \(\mathsf{SO}(3)\) in (3), is the cover given by:  
\[ \begin{array}{l}\mathbf{V}_{\pm}=\pm\!\left(\begin{array}{cc}\alpha&\beta \\ \beta^{*}&\alpha^{*}\end{array}\right)\hskip 56.905512 pt\text{where}\;\det \mathbf{V}_{\pm}=|\alpha|^{2}-|\beta|^{2}=1,\\ \stackrel{{ 2\text{.1}}}{{\longleftrightarrow}}\mathbf{P}= \mathbf{D}^{2\times 1}(\mathbf{V}_{\pm})=\left(\begin{array}{cc} \operatorname{Re}\left(\alpha^{2}\!-\!\beta^{2}\right)&\operatorname{Im}\left( \alpha^{2}\!+\!\beta^{2}\right)&2\operatorname{Im}\left(\alpha\beta\right)\\ \operatorname{Im}\left(\beta^{2}\!-\!\alpha^{2}\right)&\operatorname{Re}\left( \alpha^{2}\!+\!\beta^{2}\right)&2\operatorname{Re}\left(\alpha\beta\right)\\ -2\operatorname{Im}\left(\alpha\beta^{*}\right)&2\operatorname{Re}\left( \alpha\beta^{*}\right)&|\alpha|^{2}\!+\!|\beta|^{2}\end{array}\right).\end{array}  \tag{5}\]  
The \(\mathbf{V}\)'s form a group of matrices that satisfy \(\mathbf{V}\mathbf{O}^{1\times 1}\mathbf{V}^{\dagger}=\mathbf{O}^{1\times 1}\), with \(\mathbf{O}^{1\times 1}:=\left( \begin{smallmatrix}1&0\\ 0&-1\end{smallmatrix} \right)\), thus called special (\(\det\mathbf{V}=+1\)) _pseudo_-unitary and denoted \(\mathsf{SU}(1,1)\), while the \(\mathbf{P}\)'s also form a group of real matrices that satisfy \(\mathbf{P}\mathbf{O}^{2\times 1}\mathbf{P}^{\top}=\mathbf{O}^{2\times 1}\), where \(\mathbf{O}^{2\times 1}:=\operatorname{diag}\left(1,\,1,\,-1\right)\) characterizes the '\(2\!+\!1\)' special _pseudo_-orthogonal or _Lorentz_ group of three-dimensional relativity, denoted \(\mathsf{SO}(2,1)\). In this case, the \(3\times 3\) matrices apply to 3-vectors \(\vec{v}=(x\,y\,t)^{\top}\), and respect the'space-time' magnitude \(\vec{v}^{\top}\mathbf{O}^{2\times 1}\vec{v}\). To examine their manifolds, we again write the entries of \(\mathbf{V}\) with \(\alpha=x_{1}+\mathrm{i}x_{2}\) and \(\beta=x_{3}+\mathrm{i}x_{4}\), so that now the determinant is \(\det\mathbf{V}=x_{1}^{2}+x_{2}^{2}-x_{3}^{2}-x_{4}^{2}=1\). Written as \(x_{1}^{2}+x_{2}^{2}-x_{3}^{2}=1+x_{4}^{2}\geq 1\) in three-dimensional space, this is the surface (for \(x_{4}=0\)) andtwice the exterior (for \(|\pm x_{4}|>0\)) of a one-sheeted hyperboloid. We have thus an \(R^{3}\) manifold with a hyperbolic 'hole', which is _multiply_ connected: Trajectories that wind a different number of loops around the hole are distinct, and cannot be deformed one onto another. The map (5) is 2:1, but leaves the \(\mathsf{SU}(1,1)\) group to cover further manifolds where an angular coordinate around the hyperboloid axis can be counted any times \(2\pi\) before returning to the initial value. While the \(\mathsf{SU}(2,\mathsf{C})\) matrices do not have a real form alternative to the complex (3), we note that the complex \(\mathsf{SU}(1,1)\) matrices \(\mathbf{V}\) in (5) are equivalent to the set of real \(2\times 2\) matrices \(\mathbf{M}=\left( \begin{smallmatrix}a&b\\ c&d\end{smallmatrix} \right)\) of unit determinant \(ad-bc=1\), denoted \(\mathsf{SL}(2,R)\). Indeed, with the unitary matrix \(\mathbf{W}:=\frac{1}{\sqrt{2}}\left( \begin{smallmatrix}1&-\mathrm{i}\\ 1&\mathrm{i}\end{smallmatrix} \right)=\mathbf{W}^{-1\,\intercal}\), we have:  
\[\mathbf{M} = \left( \begin{smallmatrix}a&b\\ c&d\end{smallmatrix} \right)=\mathbf{W}^{-1}\mathbf{V}\,\mathbf{W}=\left( \begin{smallmatrix}\mathrm{Re}\left(\alpha+\beta\right)&-\mathrm{Im}\left( \alpha-\beta\right)\\ \mathrm{Im}\left(\alpha+\beta\right)&\mathrm{Re}\left(\alpha-\beta\right) \end{smallmatrix} \right), \tag{6}\] \[\mathbf{V} = \left( \begin{smallmatrix}\alpha&\beta\\ \beta^{*}&\alpha^{*}\end{smallmatrix} \right)=\mathbf{W}\,\mathbf{M}\,\mathbf{W }^{-1}=\frac{1}{2}\left( \begin{pmatrix}(a+d)-\mathrm{i}(b-c)&(a-d)+\mathrm{i}( b+c)\\ (a-d)-\mathrm{i}(b+c)&(a+d)+\mathrm{i}(b-c)\end{pmatrix} . \tag{7}\]  
This brings in the realization that finding coverings is related to finding coordinates appropriate to the task.