I'm trying to evaluate for the expectation value of the $\hat{S}_z$ operator on the $|0>_z$ eigenstate, which I'm assuming can be expressed $<\hat{S}>=<0|\hat{S}|0>$. I know the general eigenvalue equation tells me that $\hat{S} _z|0> = \hbar 0 |0>$, but this doesn't make sense when evaluating in matrix form, the output vector is $\overrightarrow{0}$:
$\hbar\ [[1,0,0],[0,0,0],[0,0,-1]]\ \times[0,1,0]\ = \hbar\ [0,0,0]$
Am I missing something or is there some reason that the expression doesn't translate to matrix form? And what is the final result for $<\hat{S}>$?
