symmetrize

Symmetrize is a verb meaning to transform an object so that it is symmetric. In mathematics and related fields, symmetrization is often achieved by combining an object with a mirror image or transpose, and then averaging the two.

In linear algebra, the standard symmetrization of a square matrix A is S = (A + A^T)/2. The result S is real and symmetric; its diagonal equals that of A, and S_ij = S_ji. Any square matrix can be decomposed uniquely into a symmetric part and an antisymmetric part, A = S + T with T = (A - A^T)/2.

In functional contexts, a function f on a symmetric domain can be symmetrized to produce an even or odd component. For example, the even part is f_s(x) = (f(x) + f(-x))/2, and the odd part is f_o(x) = (f(x) - f(-x))/2.

In data processing and geometry, symmetrization may refer to replacing an asymmetric object with a symmetric version, such as an adjacency matrix A for a directed graph by (A + A^T)/2 to obtain an undirected representation, or to Steiner or Schwarz symmetrization of shapes to enhance symmetry while preserving volume.

Notes: Symmetrization is a modeling tool, not a universal operation; the procedure changes properties and is used to simplify analysis, ensure compatibility with symmetry assumptions, or improve numerical behavior.