Symmetrization

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In mathematics, the notion of symmetrization is used to pass from any map to an alternating map.

Let S be a set and A an Abelian group. Given a map \alpha: S \times S \to A, α is termed a symmetric map if α(s,t) = α(t,s) for all s,t \in S.

The symmetrization of a general (not necessarily alternating) map \alpha: S \times S \to A is the map (x,y) \mapsto \alpha(x,y) + \alpha(y,x).

The symmetrization of a symmetric map is simply its double, while the symmetrization of an alternating map is zero.

The symmetrization of a bilinear map is bilinear.

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