Pauli spin matrices

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The Pauli spin matrices (named after physicist Wolfgang Ernst Pauli) are a set of unitary Hermitian matrices which form an orthogonal basis (along with the identity matrix) for the real Hilbert space of 2 × 2 Hermitian matrices and for the complex Hilbert spaces of all 2 × 2 matrices. They are usually denoted:

  0 & 1 \\
  1 & 0 
\end{pmatrix}, \quad
  0 & -\mathit{i} \\
  \mathit{i} & 0 
\end{pmatrix}, \quad
  1 & 0 \\
  0 & -1 

[edit] Algebraic properties


For i = 1, 2, 3:

\mbox{eigenvalues}=\pm 1\,

[edit] Commutation relations

\sigma_1\sigma_2 = i\sigma_3\,\!
\sigma_3\sigma_1 = i\sigma_2\,\!
\sigma_2\sigma_3 = i\sigma_1\,\!
\sigma_i\sigma_j = -\sigma_j\sigma_i\mbox{ for }i\ne j\,\!

The Pauli matrices obey the following commutation and anticommutation relations:

[\sigma_i, \sigma_j]     &=& 2 i\,\varepsilon_{i j k}\,\sigma_k \\[1ex]
\{\sigma_i, \sigma_j\} &=& 2 \delta_{i j} \cdot I
where \varepsilon_{ijk} is the Levi-Civita symbol, δij is the Kronecker delta, and I is the identity matrix.

The above two relations can be summarized as:

\sigma_i \sigma_j = \delta_{ij} \cdot I + i \varepsilon_{ijk} \sigma_k. \,
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