In mathematics, symmetrization is a process that converts any function in formula_1 variables to a symmetric function in formula_1 variables.
Similarly, antisymmetrization converts any function in formula_1 variables into an antisymmetric function.
Two variables.
Let formula_4 be a set and formula_5 be an additive abelian group. A map formula_6 is called a ;templatestyles src="Template:Visible anchor/styles.css" / ;symmetric map if
formula_7
It is called an ;templatestyles src="Template:Visible anchor/styles.css" / ;antisymmetric map if instead 
formula_8
The ;templatestyles src="Template:Visible anchor/styles.css" / ;symmetrization of a map formula_6 is the map formula_10
Similarly, the ;templatestyles src="Template:Visible anchor/styles.css" / ;antisymmetrization or ;templatestyles src="Template:Visible anchor/styles.css" / ;skew-symmetrization of a map formula_6 is the map formula_12
The sum of the symmetrization and the antisymmetrization of a map formula_13 is formula_14
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.
Bilinear forms.
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 formula_15 a function is skew-symmetric if and only if it is symmetric (as formula_16). 
This leads to the notion of ε-quadratic forms and ε-symmetric forms.
Representation theory.
In terms of representation theory:
As the symmetric group of order two equals the cyclic group of order two (formula_17), this corresponds to the discrete Fourier transform of order two.
"n" variables.
More generally, given a function in formula_1 variables, one can symmetrize by taking the sum over all formula_19 permutations of the variables, or antisymmetrize by taking the sum over all formula_20 even permutations and subtracting the sum over all formula_20 odd permutations (except that when formula_22 the only permutation is even).
Here symmetrizing a symmetric function multiplies by formula_19 – thus if formula_19 is invertible, such as when working over a field of characteristic formula_25 or formula_26 then these yield projections when divided by formula_27
In terms of representation theory, these only yield the subrepresentations corresponding to the trivial and sign representation, but for formula_28 there are others – see representation theory of the symmetric group and symmetric polynomials.
Bootstrapping.
Given a function in formula_29 variables, one can obtain a symmetric function in formula_1 variables by taking the sum over formula_29-element subsets of the variables. In statistics, this is referred to as bootstrapping, and the associated statistics are called U-statistics.
Notes.
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