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Pauli matrix
$$ \hat{\sigma}^x =\begin{bmatrix} && 1 \\ 1 &&\end{bmatrix} \\
\hat{\sigma}^y =\begin{bmatrix} && i \\ -i &&\end{bmatrix} \\
\hat{\sigma}^z =\begin{bmatrix} -1&& \\ && 1\end{bmatrix} $$
The rotation matrix generated by Pauli matrix
$$ \hat{R}(\hat{a},\theta) =\exp\left(-\frac{i}{2} \vec{a}\cdot\vec{\sigma}\theta\right) = \begin{bmatrix} \cos\frac{\theta}{2} +ia_z\sin\frac{\theta}{2} && (-ia_x+a_y)\sin\frac{\theta}{2}\\ (-ia_x-a_y) \sin\frac{\theta}{2}&& \cos\frac{\theta}{2} - i a_z \sin\frac{\theta}{2}\end{bmatrix} $$
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The spin angular momentum operator $\hat{\vec{S}}$ of the spin-1/2 system can be described by Pauli operator$\{\hat{\vec{\sigma}}\}$,
$$ \hat{\vec{S}} = \frac{\hbar}{2} \hat{\vec{\sigma}} $$
Three Pauli operators are described by $2\times2$-matrix form under the spin-1/2 basis $\{\left|\downarrow\right>, \left|\uparrow\right>\}$. In the language of the quantum information, spin-1/2 system is interpreted as qubit system, $\{\left|0\right>, \left|1\right>\}$. I choose this qubit representation.
$$ \hat{\sigma}^x = \left|0\right>\left<1\right| + \left|1\right>\left<0\right| =\begin{bmatrix} && 1 \\ 1 &&\end{bmatrix} \\
\hat{\sigma}^y = i\left|0\right>\left<1\right| -i \left|1\right>\left<0\right| =\begin{bmatrix} && i \\ -i &&\end{bmatrix} \\
\hat{\sigma}^z = -\left|0\right>\left<0\right| + \left|1\right>\left<1\right| =\begin{bmatrix} -1&& \\ && 1\end{bmatrix} $$
One of the important character of the Pauli operator is that the square of the each matrix is identity
$$ (\hat{\sigma}^x)^2=(\hat{\sigma}^y)^2=(\hat{\sigma}^z)^2 = \hat{I} $$
These three operators follow the commutation relation
$$ [\hat{\sigma}^j, \hat{\sigma}^k] = 2i\epsilon_{jkl}\hat{\sigma}^l $$
where $\epsilon_{jkl}$ is the Levi-Civita symbol.
The combination of the $\hat{\sigma}^x$ and $\hat{\sigma}^y$ operators generate ladder (creation-and-annihilation) operators.
$$ \hat{\sigma}^+ = \frac{1}{2}(\hat{\sigma}^x+i\hat{\sigma}^y) = \left|1\right>\left<0\right| = \begin{bmatrix} && \\ 1 &&\end{bmatrix} \\
\hat{\sigma}^- = \frac{1}{2}(\hat{\sigma}^x-i\hat{\sigma}^y) = \left|0\right>\left<1\right| = \begin{bmatrix} && 1\\ &&\end{bmatrix} $$
One of the useful operator is the number operator $\hat{n}$,
$$ \hat{n} = \hat{\sigma}^+\hat{\sigma}^- = \left|1\right>\left<1\right| = \begin{bmatrix} && \\ && 1\end{bmatrix} $$
It represent the number of the spin-up arranged spin; if the state arranged as spin-down, $n=0$, and the state arranged as spin-up, $n=1$. The relation between Pauli z operator and number operator is
$$ \hat{n} = \frac{\hat{\sigma}^z+\hat{I}}{2} $$
The rotation operator generated by spin-1/2 angular momentum $\vec{S}$ is