Nonsymmetry

Nonsymmetry refers to the absence or breaking of symmetry in a system, object, or description. It denotes that the system is not invariant under one or more symmetry operations, such as reflection, rotation, time reversal, or permutation. Nonsymmetry is a broad concept used across science and mathematics to describe directional dependence, irregularity, or unequal treatment of components within a structure.

In mathematics, nonsymmetry is often discussed in relation to matrices and tensors. A matrix is nonsymmetric if it is not equal to its transpose (A ≠ A^T). Such matrices may have nonreal eigenvalues, and they are not generally diagonalizable by orthogonal transformations; many are analyzed via Jordan forms or singular value decompositions. In geometry, objects that lack mirror or rotational symmetry are described as nonsymmetric.

In physics and chemistry, nonsymmetry appears in various contexts. Chiral molecules lack reflection symmetry, giving rise to enantiomers with distinct properties. Anisotropic materials show direction-dependent physical properties, a form of spatial nonsymmetry. In field theory and condensed matter, symmetry breaking describes a state in which a symmetric set of laws leads to a less symmetric configuration, with important consequences for dynamics and phase transitions.

Impact and applications include altered spectra and transport properties, non-reciprocal effects, and changes in selection rules. In numerical analysis, nonsymmetric problems require different algorithms and considerations compared with symmetric cases.

See also: asymmetry, symmetry breaking, anisotropy, non-symmetric matrix.