antisymmetric

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English[edit]

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Etymology[edit]

anti- +‎ symmetric.

Adjective[edit]

antisymmetric (not comparable)

  1. (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, yS, 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 'x divides y' for all x, y ∈ N.
      This relation is an antisymmetric relation on N.
  2. (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., MT = −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, and . 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., Tijk = -Tjik);
      • 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:
            ,          (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 to . []
        Exercise 21 Show that every antisymmetric bilinear form on is a wedge product of two covectors.

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