Becaure for a given J² eigenvalue there are (2j+1) different J_z eigenvalues, any linear operator which does not change J² can be represented by a (2j+1)x(2j+1) matrix. Since J_z|j,m> = hbar m |j,m>, J_z is a diagonal matrix. Since J_+/-|j,m> is proportional to |j,m +/- 1>, J_+/- have nonzero elements directly above or below the diagonal. J_x and J_y may be expressed in terms of J_+/-. Thus, all the angular momement operators for a given J² eigenvalue of hbar j(j+1), may be represented as (2j+1)x(2j+1) matrices. For the case of j=1/2, these matrices are (hbar/2) times the Pauli matrices.