Pauli Spin Matrices

$${\sigma }_x=\left[\begin{array}{cc}0,& 1\\ 1& 0\end{array}\right],{\sigma }_y=\left[\begin{array}{cc}0& -i\\ i& 0\end{array}\right],{\sigma }_z=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$$

${\hat{S}}_i=\frac{\hslash {\sigma }_i}{2}$

• I know this is true for $i=3$ or $z$, need to check this later

$${\lambda }_x=\left\{-1,1\right\},{\mathfrak{B}}_x=\frac{1}{\sqrt{2}}\left\{\left[\begin{array}{c}1\\ -1\end{array}\right],\left[\begin{array}{c}1\\ 1\end{array}\right]\right\}$$

• Eigensystem for $x$ spinor

$${\lambda }_y=\left\{-1,1\right\},{\mathfrak{B}}_y=\frac{1}{\sqrt{2}}\left\{\left[\begin{array}{c}i\\ 1\end{array}\right],\left[\begin{array}{c}-i\\ 1\end{array}\right]\right\}$$

• Eigensystem for $y$ spinor

$${\lambda }_z=\left\{-1,1\right\},{\mathfrak{B}}_z=\left\{\left[\begin{array}{c}0\\ 1\end{array}\right],\left[\begin{array}{c}1\\ 0\end{array}\right]\right\}$$

• Eigensystem for $z$ spinor