Editing Clifford module (section)

==Matrix representations of real Clifford algebras==
We will need to study ''anticommuting'' [[matrix (mathematics)|matrices]] ({{nowrap|1=''AB'' = −''BA''}}) because in Clifford algebras orthogonal vectors anticommute
:<math> A \cdot B = \frac{1}{2}( AB + BA ) = 0.</math>

For the real Clifford algebra <math>\mathbb{R}_{p,q}</math>, we need {{nowrap|''p'' + ''q''}} mutually anticommuting matrices, of which ''p'' have +1 as square and ''q'' have −1 as square.
:<math> \begin{matrix}
\gamma_a^2 &=& +1 &\mbox{if} &1 \le a \le p \\
\gamma_a^2 &=& -1 &\mbox{if} &p+1 \le a \le p+q\\
\gamma_a \gamma_b &=& -\gamma_b \gamma_a &\mbox{if} &a \ne b. \ \\
\end{matrix}</math>

Such a basis of [[gamma matrices]] is not unique. One can always obtain another set of gamma matrices satisfying the same Clifford algebra by means of a similarity transformation.

:<math>\gamma_{a'} = S \gamma_{a} S^{-1} ,</math>

where ''S'' is a non-singular matrix. The sets ''γ''<sub>''a''′</sub> and ''γ''<sub>''a''</sub> belong to the same equivalence class.