Hello,
If you write out the two Pauli matrices in Dirac notation (in the computational basis |0>, |1>) and do the tensor product by expanding you find a sum of four terms. This is the 4x4 matrix written out in Dirac notation (in the computational basis |00>, |01>, |10>, |11>). You can then see that this matrix is the diagonal matrix with on the diagonal (+1, -1, -1, +1).
More about it in next class also.
Nicolas Macris
