Pauli matrices

Remarks

They satisfy $\sigma_j^2 = I$ and, for $j, k \in \set{1, 2, 3}$, $$\sigma_j \sigma_k = \delta_{jk} I + i \sum_{l=1}^{3} \varepsilon_{jkl} \sigma_l,$$ where $\delta_{jk}$ is the Kronecker delta and $\varepsilon_{jkl}$ is the Levi-Civita symbol. For example, $\sigma_x \sigma_y = i\sigma_z$, as a direct multiplication shows. Hence $[\sigma_j, \sigma_k] = 2i \sum_l \varepsilon_{jkl}\sigma_l$ and $\sigma_j\sigma_k + \sigma_k\sigma_j = 2\delta_{jk} I$.

The matrices $I, \sigma_x, \sigma_y, \sigma_z$ form a basis of the complex vector space of $2 \times 2$ matrices. They also form a basis of the real vector space of $2 \times 2$ Hermitian matrices. The matrices $i\sigma_1, i\sigma_2, i\sigma_3$ form a basis of the Lie algebra $\mathfrak{su}(2)$ of SU(2). In quantum mechanics, $\tfrac12 \sigma_j$ are the spin observables of a spin-$\tfrac12$ particle, with $\hbar = 1$. In quantum computing, $X$ is the bit flip, $Z$ the phase flip, and $Y = iXZ$.