Question

2. Which of the operators Sx, Sy, S, (if any) are Hermitian? Do the Hermitian operators have real eigenvalues, as expected?

Answer 1

$S_x=\frac {\hbar}{2}\begin{pmatrix} 0 &1 \\1 &0 \end{pmatrix}$

$S_x^{\dagger}=\frac {\hbar}{2}\begin{pmatrix} 0 &1 \\1 &0 \end{pmatrix}=S_x$

$S_y=\frac {\hbar}{2}\begin{pmatrix} 0 &-i \\i &0 \end{pmatrix}$

$S_y^{\dagger}=\frac {\hbar}{2}\begin{pmatrix} 0 &-i \\i &0 \end{pmatrix}=S_y$

$S_z=\frac {\hbar}{2}\begin{pmatrix} 1&0\\0&-1 \end{pmatrix}$

$S_z^{\dagger}=\frac {\hbar}{2}\begin{pmatrix} 1&0\\0&-1 \end{pmatrix}=S_z$

So all the above operators are Hermitian.

Yes, the Hermitian operators have real eigenvalues.