So I understand that the tensor product of two vector spaces constitutes a new vector space. In the case of spin coupling you have some arbitrary vector |x,z> where x represents the spin state of particle 1 and z represents the spin state of particle 2; bot spin states being general 2 element vectors. This is where I question my understanding, from what I can see ... the tensor product of individual eigenspaces containing vector x and z creates a new eigenspace that holds the general vector |x,z>. Also it seems that you can construct a matrix out of your eigenstates using the basis from your eigenspaces holding the spinstates x and z. So is my understanding correct? If I am right, my other question is ... how do you construct such a matrix?