Problem 2316. Spin Matrices

Yaroslav

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The spin of a particle is a fundamental property in quantum physics. We shall inspect below matrix representations of such spin operators.
Suppose you have integer or half-integer spin of value s. The matrices Sx, Sy and Sz representing it have the following properties:
Si (with i={x,y,z}) are traceless Hermitian matrices;
Commutation relations (a): [ Si,Sj ] = i εijk Sk, where [·,·] is the commutator and εijk is the Levi-Civita symbol.
Commutation relations (b): [ Si,S² ] = 0, where S² = Sx²+Sy²+Sz²;
Eigenvalues: S² = j(j+1)·I and Sz = diag( -j/2, -j/2+1, … ,j/2-1, j ), where I is the identity matrix.
See also this article for more reference.
Examples
 [Sx,Sy,Sz] = spin_matrices(1/2)
 Sx = 
     0      0.5
     0.5    0
 Sy = 
     0     -0.5i
     0.5i   0
 Sz = 
     0.5    0
     0     -0.5
Note:
The usual cheats are not allowed!