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anti- +‎ symmetric.


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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