I see that for total ket in QM of hydrogen atom we define a tensor product of kets of spatial and spin spaces, upon which spatial and spin operators, operate respectively.
For the total angular momentum operator we "add" two operators, one spatial the other one spin.
But does this make sense in a matrix representation?
I mean we are adding a multidimensional matrix (based on the principal quantum number) to a two dimensional matrix in the spin space.
Given the orbital angular momentum operator $L$ on the "spatial space" $\mathcal{H}_1$ and the spin angular momentum operator $S$ on the "spin space" $\mathcal{H}_2$, we have the total angular momentum operator on the combined space $\mathcal{H}_1\otimes\mathcal{H}_2$ given by $$ J = L \otimes \mathbf{1} + \mathbf{1} \otimes S$$ where $\mathbf{1}$ is the identity operator. This is what is meant by "adding" oribtal and spin angular momentum to total angular momentum - you first need to extend the operators to the whole space of states (by tensoring them with the "do nothing" operator, the identity, on the part where they weren't defined before), and then add these extensions.
