# 2 \(\times\) 2 Matrices: Manifolds, Realizations, Applications
## 2 \(\times\) 2 Matrices Covering 3 \(\times\) 3 Matrices
### Unitary Matrices Cover Orthogonal Matrices  
To see and prove the cover provided by spin over rotations, it is sufficient to provide a map between a subset of complex \(2\times 2\) and a subset of \(3\times 3\) orthogonal matrices:  
\[\begin{array}{l}\mathbf{U}_{\pm}=\pm\!\left(\begin{array}{cc}\alpha&\beta\\ -\beta^{*}&\alpha^{*}\end{array}\right)\hskip 56.905512 pt\text{where}\;\det \mathbf{U}_{\pm}=|\alpha|^{2}+|\beta|^{2}=1,\\ \stackrel{{ 2\text{.1}}}{{\longleftrightarrow}}\mathbf{O}= \mathbf{D}^{3\times 3}(\mathbf{U}_{\pm})=\left(\begin{array}{cc}\operatorname{Re} \left(\alpha^{2}\!-\!\beta^{2}\right)&\operatorname{Im}\left(\alpha^{2}\!+ \!\beta^{2}\right)&-2\operatorname{Re}\left(\alpha\beta\right)\\ \operatorname{Im}\left(\beta^{2}\!-\!\alpha^{2}\right)&\operatorname{Re} \left(\alpha^{2}\!+\!\beta^{2}\right)&2\operatorname{Im}\left(\alpha\beta \right)\\ 2\operatorname{Re}\left(\alpha\beta^{*}\right)&2\operatorname{Im}\left(\alpha \beta^{*}\right)&|\alpha|^{2}\!-\!|\beta|^{2}\end{array}\right).\end{array} \tag{3}\]  
Upon writing out the complex numbers \(\alpha=x_{1}+\mathrm{i}x_{2}\) and \(\beta=x_{3}+\mathrm{i}x_{4}\), we see that \(\det\mathbf{U}=\sum_{i=1}^{4}x_{i}^{2}=1\) is a 3-sphere or, if we write \(\sum_{i=1}^{3}x_{i}^{2}=1-x_{4}^{2}\leq 1\), we recognize it as the boundary (for \(x_{4}=0\)), and twice the center (for \(x_{4}=\pm 1\)) and the interior (for \(0<|\pm x_{i}|<1\)) of a 2-sphere. The 2:1 map (3) is quadratic and covers twice that set of real \(3\times 3\) matrices. The former is the set of all \(2\times 2\) unitary matrices of unit determinant \(\mathbf{U}\,\mathbf{U}^{\dagger}=\mathbf{1}\), that form the group of _special_ (\(\det\mathbf{U}=+1\)) _unitary_ matrices denoted \(\mathsf{SU}(2)\), covering 2:1 the group of _special_ (\(\det\mathbf{O}=+1\)) _orthogonal_ matrices \(\mathbf{O}\,\mathbf{O}^{\top}=\mathbf{1}\), denoted \(\mathsf{SO}(3)\). (We indicate complex conjugation by \(\xi^{*}\), matrix transposition by \(\mathbf{X}^{\top}\), and matrix adjunction by \(\mathbf{X}^{\top}:=(\mathbf{X}^{*})^{\top}\).) These matrices will apply to and rotate 3-vectors \(\vec{u}=(x\,y\,z)^{\top}\) respecting their magnitude \(|\vec{u}|^{2}=\vec{u}^{\top}\vec{u}\).  
Most importantly, the map (3) preserves the group properties between the \(2\times 2\) and \(3\times 3\) matrices:  
\[\mathbf{U}_{1}\mathbf{U}_{2}=\mathbf{U}_{3}\quad\Longrightarrow\quad\mathbf{ D}^{3\times 3}(\mathbf{U}_{1})\,\mathbf{D}^{3\times 3}(\mathbf{U}_{2})= \mathbf{D}^{3\times 3}(\mathbf{U}_{3}), \tag{4}\]  
while the unit is \(\mathbf{D}^{3\times 3}(\pm\mathbf{1})=\mathbf{1}^{3\times 3}\), all inverses \(\mathbf{D}^{3\times 3}(\pm\mathbf{U}^{-1})=\mathbf{D}^{3\times 3}(\pm \mathbf{U})^{-1}\) exist, and associativity holds as it does for all matrices.  
The \(\mathsf{SU}(2)\) manifold is a 3-sphere which is _simply_ connected, as all spheres are; this means that any closed path in the manifold can be contracted to a point. The \(\mathsf{SO}(3)\) manifold on the other hand can be visualized as the interior of a 2-sphere corresponding to all rotation axes \(\widehat{n}(\theta,\phi)\) with radii given by the rotation angles \(0\leq\psi\leq\pi\). However since a rotation by \(\pi\) around \(\widehat{n}\) is the same as a rotation by \(\pi\) around the antipodal \(-\widehat{n}\) point on the surface of that sphere, the two transformations are identified as the same group element. This manifold is thus _doubly_ connected: Trajectories which cross the sphere'surface' an even number of times can be contracted to a point; those that cross it an odd number can contract only to a single-crossing trajectory.