skewsymmetry

Skew symmetry is a property of a mathematical object that changes sign when its arguments are swapped. In linear algebra, the term is most often applied to matrices and bilinear forms.

A real square matrix A is skew-symmetric if A^T = -A. This implies that all diagonal entries are zero and that a_ij = -a_ji for all i, j. Equivalently, for vectors v and w, the bilinear form B(v,w) = v^T A w satisfies B(v,w) = -B(w,v). Skew-symmetric forms are also called antisymmetric.

Key properties include that the eigenvalues of a real skew-symmetric matrix occur in conjugate pairs ±iα and, for odd dimension, the determinant is zero. The rank of a real skew-symmetric matrix is even. There exists an orthogonal matrix Q such that Q^T A Q is block diagonal with 2x2 blocks of the form [ [0, λ_k], [-λ_k, 0] ] and possibly zero blocks.

Over fields of characteristic not 2, every skew-symmetric form is alternating: B(v,v) = 0 for all v, and every alternating bilinear form arises from a skew-symmetric form.

In a broader sense, skew-symmetry describes a 2-form ω with ω(v,w) = -ω(w,v). The matrix representation of ω is skew-symmetric, and such 2-forms are central in the exterior algebra and differential geometry.

Applications appear in physics and engineering, notably in describing infinitesimal rotations and angular momentum, as well as in computer graphics and the study of Lie algebras such as so(n). See also antisymmetric structures and exterior algebra.