Pauli spin matrices

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 &times; 2 Hermitian matrices and for the complex Hilbert spaces of all 2 &times; 2 matrices. They are usually denoted:


 * $$\sigma_x=\begin{pmatrix}

0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \sigma_y=\begin{pmatrix} 0 & -\mathit{i} \\ \mathit{i} & 0 \end{pmatrix}, \quad \sigma_z=\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$$

Algebraic properties

 * $$\sigma_x^2=\sigma_y^2=\sigma_z^2=I$$

For i = 1, 2, 3:


 * $$\mbox{det}(\sigma_i)=-1\,$$


 * $$\mbox{Tr}(\sigma_i)=0\,$$


 * $$\mbox{eigenvalues}=\pm 1\,$$

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:


 * $$\begin{matrix}

[\sigma_i, \sigma_j]    &=& 2 i\,\varepsilon_{i j k}\,\sigma_k \\[1ex] \{\sigma_i, \sigma_j\} &=& 2 \delta_{i j} \cdot I \end{matrix}$$


 * where $$\varepsilon_{ijk}$$ is the Levi-Civita symbol, $$\delta_{ij}$$ is the Kronecker delta, and I is the identity matrix.

The above two relations can be summarized as:


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