Antisymmetric

Antisymmetric describes a property in various branches of mathematics where swapping two indices, arguments, or components introduces a sign change, or where a relation forbids certain symmetric coincidences. The specifics depend on context, but the common idea is a built-in sign-reversal under permutation.

In order theory and relations, a binary relation R on a set S is antisymmetric if, whenever aRa and bRb and aRb and bRa, then a = b. More precisely, for all a, b in S, if aRb and bRa, then a = b. This excludes the possibility of two distinct elements being mutually related in both directions.

In linear algebra, a matrix A is antisymmetric (or skew-symmetric) if A^T = -A. For real matrices this implies all diagonal entries are zero and a_{ij} = -a_{ji}. Such matrices have purely imaginary eigenvalues in general, and eigenvalues occur in pairs ±iλ. The dimension of the space of antisymmetric matrices is n(n−1)/2 for n×n matrices.

In multilinear algebra, an antisymmetric or alternating tensor T changes sign when any two of its indices are swapped; equivalently, T vanishes whenever two indices coincide. These objects span the exterior algebra via wedge products, with dimensions given by binomial coefficients.

Antisymmetry appears in physics to describe fermionic wavefunctions, in geometry in exterior calculus, and in differential forms as the fundamental property of wedge products. The term is often used interchangeably with skew-symmetric in these contexts.