Cheat sheets/Spin matrices

$\sigma _{x}=\sigma _{1}={\begin{pmatrix}0&1\\1&0\end{pmatrix}}$

$\sigma _{y}=\sigma _{2}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}}$

$\sigma _{z}=\sigma _{3}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}$