The following operators are spin operators acting on the spin coordinates.
$s_x = \frac{1}{2} \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix}$, $s_y = \frac{1}{2} \begin{pmatrix} 0 & -i \ i & 0 \end{pmatrix}$, $s_z = \frac{1}{2} \begin{pmatrix} 1 & 0 \ 0 & -1 \end{pmatrix}$
a) Find the eigenvalue and eigenvector of $s_z$.
b) Are the eigenvectors from (a) orthonormal? Prove it.
c) Show that $[s_x, s_y] = is_z$