The Pauli spin matrices represent the three (x,y,z) spin operators acting on a spin 1/2 particle (such as an electron): that is to say that the spin operator can be written as S = (1/2)(h-bar)σ. They are conventionally written as:
σ1 = (0 1)
(1 0)
σ2 = (0 -i)
(i 0)
σ3 = (1 0)
(0 -1)
They are each
unitary,
hermitian and
trace-free (indeed, the form a basis of sl
2(
C)). Also, they
anticommute in pairs, and satisfy the
commutation relation [ σ
i,σ
j ] = 2iε
ijkσ
k.