Quantum Pauli matrices

\[\newcommand{\im}{\mathrm{i}\mkern1mu}\]

The Pauli matrices are defined as

\[\begin{equation} X = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix},~~Y = \begin{bmatrix} 0 & -\im \\ \im & 0 \end{bmatrix},~~Z=\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}. \end{equation}\]

Each of them defines a transformation of a qbit state. For instance,

\[\begin{equation} X \ket{0} = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \end{bmatrix} = \ket{1}. \end{equation}\]

Find more information on Wikipedia.