Các ma trận Pauli có tính giao hoán. Cụ thể:

[ σ a , σ b ] = 2 i ε a b c σ c , {\displaystyle [\sigma _{a},\sigma _{b}]=2i\varepsilon _{abc}\,\sigma _{c}\,,}

và phi giao hoán:

{ σ a , σ b } = 2 δ a b I . {\displaystyle \{\sigma _{a},\sigma _{b}\}=2\delta _{ab}\,I.}

Xét các ví dụ sau:

[ σ 1 , σ 2 ] = 2 i σ 3 [ σ 2 , σ 3 ] = 2 i σ 1 [ σ 3 , σ 1 ] = 2 i σ 2 [ σ 1 , σ 1 ] = 0 { σ 1 , σ 1 } = 2 I { σ 1 , σ 2 } = 0 . {\displaystyle {\begin{aligned}\left[\sigma _{1},\sigma _{2}\right]&=2i\sigma _{3}\,\\\left[\sigma _{2},\sigma _{3}\right]&=2i\sigma _{1}\,\\\left[\sigma _{3},\sigma _{1}\right]&=2i\sigma _{2}\,\\\left[\sigma _{1},\sigma _{1}\right]&=0\,\\\left\{\sigma _{1},\sigma _{1}\right\}&=2I\,\\\left\{\sigma _{1},\sigma _{2}\right\}&=0\,.\\\end{aligned}}}