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A term's definition may require additional properties that are not listed in this table. In mathematics , a binary relation R {\displaystyle R} on a set X {\displaystyle X} is antisymmetric if there is no pair of distinct elements of X {\displaystyle X} each of which is related by R {\displaystyle R} to the other.
A tensor A that is antisymmetric on indices and has the property that the contraction with a tensor B that is symmetric on indices and is identically 0. For a general tensor U with components U i j k … {\displaystyle U_{ijk\dots }} and a pair of indices i {\displaystyle i} and j , {\displaystyle j,} U has symmetric and antisymmetric parts ...
In two dimensions, the Levi-Civita symbol is defined by: = {+ (,) = (,) (,) = (,) = The values can be arranged into a 2 × 2 antisymmetric matrix: = (). Use of the two-dimensional symbol is common in condensed matter, and in certain specialized high-energy topics like supersymmetry [1] and twistor theory, [2] where it appears in the context of 2-spinors.
Skew-symmetric graph; Self-complementary graph; In mathematics, especially linear algebra, and in theoretical physics, the adjective antisymmetric (or skew-symmetric) is used for matrices, tensors, and other objects that change sign if an appropriate operation (e.g. matrix transposition) is performed. See:
Since this definition is independent of the choice of basis, skew-symmetry is a property that depends only on the linear operator and a choice of inner product. skew symmetric matrices can be used to represent cross products as matrix multiplications.
The wedge product is the anti-symmetric form of the ⊗ operation. The quotient space of T(V) on which it becomes an internal operation is the exterior algebra of V; it is a graded algebra, with the graded piece of weight k being called the k-th exterior power of V. Symmetric power, symmetric algebra
The symmetrization and antisymmetrization of a bilinear map are bilinear; thus away from 2, every bilinear form is a sum of a symmetric form and a skew-symmetric form, and there is no difference between a symmetric form and a quadratic form. At 2, not every form can be decomposed into a symmetric form and a skew-symmetric form.
A square matrix with all entries off the anti-diagonal equal to zero. Anti-Hermitian matrix: Synonym for skew-Hermitian matrix. Anti-symmetric matrix: Synonym for skew-symmetric matrix. Arrowhead matrix: A square matrix containing zeros in all entries except for the first row, first column, and main diagonal. Band matrix