Matrix operations on Quantum States in a composite quantum system Intro (you may skip this if you're an expert, I'm including this for completeness):
Say I have two bases for two systems,
The first is a spin-1/2 system  $|+\rangle = \left(\begin{array}{c}
1\\0
\end{array}\right),|-\rangle=\left(\begin{array}{c}
0\\1
\end{array}\right)$
The second is a spin-1 system, with $|1_+\rangle=\left(\begin{array}{c}
1\\0\\0
\end{array}\right),|1_0\rangle=\left(\begin{array}{c}
0\\1\\0
\end{array}\right),|1_-\rangle=\left(\begin{array}{c}
0\\0\\1
\end{array}\right)$
Now for the first system, I can use the Pauli matrix 
$$\hat{S_z}=\frac{1}{2}\hbar\hat{\sigma}_z = \left(
\begin{array}{cc}
 -\frac{\hbar }{2} & 0 \\
 0 & \frac{\hbar }{2} \\
\end{array}
\right)$$
in order to get the projection of my state on the z-axis. Likewise, I could use the projection matrix
$$\hat{J}_z=\hbar\left(
\begin{array}{ccc}
 1  & 0 & 0 \\
 0 & 0 & 0 \\
 0 & 0 & -1  \\
\end{array}
\right)$$
To project the other state on the z-axis. Those operators will act on my basis in the following way:
$$\hat{S}_z|+\rangle=\frac{\hbar}{2}|+\rangle$$
$$\hat{J}_z|1_+\rangle=\hbar|1_+\rangle$$
Problem: (here comes the question)
So far everything is good! Now the problem comes when I introduce a space for the composite system, so I'm getting the basis
$$|S_z\rangle\otimes |J_z\rangle\rightarrow\left\{ |+,1_+\rangle,|+,1_0\rangle,|+,1_-\rangle,|-,1_+\rangle,|-,1_0\rangle,|-,1_-\rangle\right\}$$
Now the question is: how do I use the matrix formalism to have such operations just like I had them before in the single systems:
$$S_z|+,1_+\rangle=\frac{\hbar}{2}|+,1_+\rangle$$
$$S_z|+,1_-\rangle=\frac{\hbar}{2}|+,1_-\rangle$$
$$J_z|+,1_0\rangle=0|+,1_0\rangle$$
$$J_z|+,1_-\rangle=-\hbar|+,1_-\rangle$$
In other words, how do I write the state-kets and the operators in the composite system in matrix formalism (just like I showed in the beginning) to give results compatible with what I would expect in the examples?
Is this wrong in some way?
Every time I try to do this with Kronecker Product (like $\hat{S}_z \otimes \hat{J}_z$) I arrive at a mess, and I get terms proportional to $\hbar^2$, and I don't get the eigen-values I expect, and I'm not sure what I'm doing wrong. Could you please show me how to do this?
Thank you.
 A: Just a remark on the introduction. $\hat{S}_z$ does not "project" states on the $z$ direction in spin space. Indeed $\hat{S}_z ^2 = \hbar/2 \;\text{Id}_{2\times 2}\neq \hat{S}_z$, while a projector $P$ verifies $P^2=P$. Actually, $\hat{S}_z$ rotates states around the $z$ axis in spin space ($e^{i\theta \hat{S}_z}$ rotates states by an angle $\theta$ around the $z$ axis).
Back to the main problem, I have two equivalent answers


*

*The matrix elements of $\hat{S}_z\otimes \hat{J}_z$ are given by
\begin{equation}
\langle \pm , 1_{+,0,-}|\hat{S}_z\otimes \hat{J}_z |\pm ', 1'_{+,0,-}\rangle =\langle \pm| \hat{S}_z|\pm '\rangle \langle 1_{+,0,-}|\hat{J}_z |1'_{+,0,-}\rangle
\end{equation}
where $\pm$, and $\pm'$ refers to the eigenstates of $S_z$, while $1_{+,0,-}$  and $1'_{+,0,-}$ are the eigenstates of $J_z$.
Hence you can compute all the matrix elements separately. For instance, in your basis the top left element of the matrix is
\begin{equation}
\langle + , 1_+|\hat{S}_z\otimes \hat{J}_z |+, 1_+\rangle =\langle +| \hat{S}_z|+ \rangle \langle 1_+|\hat{J}_z |1_+\rangle = \frac{\hbar}{2}\times \hbar = \frac{1}{2}\hbar^2
\end{equation}
and the other elements have to be computed in the same way (most of them are zero).

*Here you deal with direct product operators, since $\hat{S}_\alpha$ does not act on $\hat{J}_\beta$ eigenstates (whatever $\alpha$ and $\beta$) and vice versa. Thus, you get block diagonal matrices like 
\begin{equation}
 \hat{S}_z \otimes \hat{J}_z = \begin{pmatrix} \frac{\hbar}{2}J_z & 0_{3\times 3} \\ 0_{3\times 3} & -\frac{\hbar}{2}J_z \end{pmatrix}\quad .
\end{equation}
where $0_{3\times 3}$ is the 3 by 3 matrix with all elements set to zero and $J_z$ is the 3 by 3 matrix given by
\begin{equation}
J_z = \hbar \begin{pmatrix} 1 & 0 & 0 \\ 0 &0  & 0\\ 0 & 0 & -1 \end{pmatrix}
\end{equation}
in your basis.
The other matrices your are interested in, like $S_x\otimes J_y$ (as far as I understood) can be similarly obtained by one of those two ways.
Hope this helps!
