Symmetrized

Symmetrized is the past participle of symmetrize. In mathematics, physics and related fields, it denotes making an object invariant under a symmetry or exchange of variables by averaging over the symmetry group. If a group G acts on a set X and f is defined on X, the symmetrized form f_sym is typically f_sym(x) = (1/|G|) sum_{g in G} f(g·x).

In linear algebra, symmetrization produces a symmetric structure. The symmetrized part of a square matrix A is (A + A^T)/2. This is the maximal symmetric component in the standard decomposition A = S + N with S = (A + A^T)/2 and N = (A - A^T)/2, where N is skew-symmetric.

In tensor analysis, symmetrization extends to tensors by averaging over permutations of indices. The symmetrized tensor T_sym(i1...in) = (1/n!) sum_{π in S_n} T_{i_{π(1)}... i_{π(n)}} is invariant under permutations of those indices. Partial symmetries can be imposed with more elaborate symmetrizers, particularly in representation theory and differential geometry.

In probability, symmetrization is a technique used to bound distributions of sums, often by introducing Rademacher variables ε_i and forming the symmetrized sum ∑ ε_i X_i. This helps compare empirical processes to their symmetrized counterparts.

In quantum mechanics and related fields, the symmetrized (or Weyl) product of noncommuting operators A and B is (AB + BA)/2, yielding a Hermitian observable. Symmetrization also appears in representation theory as a symmetrizer projecting onto symmetric subspaces.

The term emphasizes invariance and is widely used in algebra, geometry, analysis, and physics to produce objects with desired symmetry properties.