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 × 2 Hermitian matrices and for the complex Hilbert spaces of all 2 × 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

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

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

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

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

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:

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