In mathematics, symmetrization is a process that converts any function in ${\displaystyle n}$ variables to a symmetric function in ${\displaystyle n}$ variables. Similarly, antisymmetrization converts any function in ${\displaystyle n}$ variables into an antisymmetric function.

Let ${\displaystyle S}$ be a set and ${\displaystyle A}$ be an additive abelian group. A map ${\displaystyle \alpha :S\times S\to A}$ is called a symmetric map if $${\displaystyle \alpha (s,t)=\alpha (t,s)\quad {\text{ for all }}s,t\in S.}$$ It is called an antisymmetric map if instead $${\displaystyle \alpha (s,t)=-\alpha (t,s)\quad {\text{ for all }}s,t\in S.}$$

The symmetrization of a map ${\displaystyle \alpha :S\times S\to A}$ is the map ${\displaystyle (x,y)\mapsto \alpha (x,y)+\alpha (y,x).}$ Similarly, the antisymmetrization or skew-symmetrization of a map ${\displaystyle \alpha :S\times S\to A}$ is the map ${\displaystyle (x,y)\mapsto \alpha (x,y)-\alpha (y,x).}$

The sum of the symmetrization and the antisymmetrization of a map ${\displaystyle \alpha }$ is ${\displaystyle 2\alpha .}$ Thus, away from 2, meaning if 2 is invertible, such as for the real numbers, one can divide by 2 and express every function as a sum of a symmetric function and an anti-symmetric function.

The symmetrization of a symmetric map is its double, while the symmetrization of an alternating map is zero; similarly, the antisymmetrization of a symmetric map is zero, while the antisymmetrization of an anti-symmetric map is its double.

The symmetrization and antisymmetrization of a bilinear map are bilinear; thus away from 2, every bilinear form is a sum of a symmetric form and a skew-symmetric form, and there is no difference between a symmetric form and a quadratic form.

At 2, not every form can be decomposed into a symmetric form and a skew-symmetric form. For instance, over the integers, the associated symmetric form (over the rationals) may take half-integer values, while over ${\displaystyle \mathbb {Z} /2\mathbb {Z} ,}$ a function is skew-symmetric if and only if it is symmetric (as ${\displaystyle 1=-1}$).

This leads to the notion of ε-quadratic forms and ε-symmetric forms.