antisymmetric

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antisymmetric

Wikipedia

Etymology

From anti- +‎ symmetric.

Adjective

antisymmetric (not comparable)

(set theory, order theory, of a binary relation R on a set S) Having the property that, for any two distinct elements of S, at least one is not related to the other via R; equivalently, having the property that, for any x, y ∈ S, if both xRy and yRx then x=y.
- 1987, David C. Buchthal, Douglas E. Cameron, Modern Abstract Algebra, Prindle, Weber & Schmidt, page 479,
  The standard example for an antisymmetric relation is the relation less than or equal to on the real number system.
- 2006, S. C. Sharma, Metric Space, Discovery Publishing House, page 73:
  (i) The identity relation on a set A is an antisymmetric relation.
  (ii) Let R be a relation on the set N of natural numbers defined by
  x R y $\Leftrightarrow$ 'x divides y' for all x, y ∈ N.
  This relation is an antisymmetric relation on N.
(linear algebra, of certain mathematical objects) Whose sign changes on the application of a matrix transpose or some generalisation thereof:
1. (of a matrix) Whose transpose equals its negative (i.e., M^T = −M);
  - 1974, Robert McCredie May, Stability and Complexity in Model Ecosystems, Princeton University Press, page 193:
    The eigenvalues of an antisymmetric matrix are all purely imaginary numbers, and occur as conjugate pairs, $+iw$ and $-iw$ . As a corollary it follows that an antisymmetric matrix of odd order necessarily has one eigenvalue equal to zero; antisymmetric matrices of odd order are singular.
2. (of a tensor) That changes sign when any two indices are interchanged (e.g., T_ijk = -T_jik);
  - 1986, Millard F. Beatty Jr., Principles of Engineering Mechanics, Volume 1: Kinematics - The Geometry of Motion, Plenum Press, page 163:
    Notice that the tensors defined by:
    $\textstyle T_{S}\equiv {\frac {1}{2}}(T+T^{T})$ , $\textstyle T_{A}\equiv {\frac {1}{2}}(T-T^{T}),$ (3.47)
    are the symmetric and antisymmetric parts, respectively; they are known as the symmetric and antisymmetric parts of T.
3. (of a bilinear form) For which B(w,v) = -B(v,w).
  - 2012, Stephanie Frank Singer, Symmetry in Mechanics: A Gentle, Modern Introduction, Springer, page 28:
    Antisymmetric bilinear forms and wedge products are defined exactly as above, only now they are functions from $\mathbb {R} ^{n}\times \mathbb {R} ^{n}$ to $\mathbb {R}$ . […]
    Exercise 21 Show that every antisymmetric bilinear form on $\mathbb {R} ^{3}$ is a wedge product of two covectors.