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

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

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

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_i\sigma_j = -\sigma_j\sigma_i\mbox{ for }i\ne j\,\!}

The Pauli matrices obey the following commutation and anticommutation relations:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta_{ij}} is the Kronecker delta, and I is the identity matrix.

The above two relations can be summarized as:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_i \sigma_j = \delta_{ij} \cdot I + i \varepsilon_{ijk} \sigma_k \,} .