The characteristic equation for the spin operator S𝓏 is

    S𝓏 (S𝓏 + ħ)(S𝓏 - ħ) = 0.

The eigenvalues, which are the roots of this equation, are

    λ = 0, ±ħ.

The S𝓏 operator is already diagonalized in its own basis, so the matrix has the immediately constructable form of

    S𝓏 ≐
            ⎛ 1 0 0  ⎞
          ħ ⎜ 0 0 0  ⎟
            ⎝ 0 0 -1 ⎠


To produce the operator S𝓏, one can apply the rotation matrix for a rotation about the y axis., where the angle of rotation is π/2.

            ⎛ cosθ  0  sinθ ⎞   ⎛ 0  0  1 ⎞   
        R = ⎜   0   1   0   ⎟ = ⎜ 0  1  0 ⎟  
            ⎝-sinθ  0  cosθ ⎠   ⎝-1  0  0 ⎠

S𝓍 = S𝓏 R ≐ 
           ⎛ 1 0 0  ⎞ ⎛ 0  0  1 ⎞   ⎛ 0  0  1 ⎞
         ħ ⎜ 0 0 0  ⎟ ⎜ 0  1  0 ⎟ = ⎜ 0  1  0 ⎟
           ⎝ 0 0 -1 ⎠ ⎝-1  0  0 ⎠   ⎝-1  0  0 ⎠





The same is true for the S𝓍 operator in its basis. To express this operator in the z basis, however, it must be diagonalized.