Pauli Matrix

linear-algebra

Definition

Pauli Matrix

A Pauli matrix is one of the three $2\times 2$ matrices

$${\sigma }_x=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right],{\sigma }_y=\left[\begin{array}{cc}0& -i\\ i& 0\end{array}\right],{\sigma }_z=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right].$$

They act as linear operators on ${\mathbb{C}}^2$.

Properties

The Pauli matrices are Hermitian and unitary.

They satisfy:

$${\sigma }_x^2={\sigma }_y^2={\sigma }_z^2=I$$

and they pairwise anticommute:

$${\sigma }_i{\sigma }_j=-{\sigma }_j{\sigma }_i(i\ne j).$$