Symmetrization

In mathematics, the notion of symmetrization is used to pass from any map to a symmmetric map.

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

The symmetrization of a general (not necessarily alternating) map $alpha:\; S\; imes\; S\; o\; A$ 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.