Pauli spin matrices

The Pauli spin matrices are a set of unitary Hermitian matrices which form an orthogonal basis (along with the identity matrix) for the real Hilbert space of 2x2 Hermitian matrices and for the complex Hilbert spaces of all 2x2 matrices. They are usually denoted:
$\sigma _{x}={\begin{pmatrix}0&1\\1&0\end{pmatrix}},\sigma _{y}={\begin{pmatrix}0&-{\mathit {i}}\\{\mathit {i}}&0\end{pmatrix}},\sigma _{z}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}$

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

$\sigma _{1}\sigma _{2}=i\sigma _{3}\,\!$

$\sigma _{3}\sigma _{1}=i\sigma _{2}\,\!$

$\sigma _{2}\sigma _{3}=i\sigma _{1}\,\!$

$\sigma _{i}\sigma _{j}=-\sigma _{j}\sigma _{i}{\mbox{ for }}i\neq j\,\!$

The Pauli matrices obey the following commutation and anticommutation relations:

${\begin{matrix}[\sigma _{i},\sigma _{j}]&=&2i\,\varepsilon _{ijk}\,\sigma _{k}\\[1ex]\{\sigma _{i},\sigma _{j}\}&=&2\delta _{ij}\cdot I\end{matrix}}$

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:

$\sigma _{i}\sigma _{j}=\delta _{ij}\cdot I+i\varepsilon _{ijk}\sigma _{k}\,$.