# antisymmetric

## English

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### Etymology

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 ${\displaystyle \Leftrightarrow }$  '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, ${\displaystyle +iw}$  and ${\displaystyle -iw}$ . 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., 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:
${\displaystyle \textstyle T_{S}\equiv {\frac {1}{2}}(T+T^{T})}$ ,     ${\displaystyle \textstyle T_{A}\equiv {\frac {1}{2}}(T-T^{T}),}$      (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 ${\displaystyle \mathbb {R} ^{n}\times \mathbb {R} ^{n}}$  to ${\displaystyle \mathbb {R} }$ . []
Exercise 21 Show that every antisymmetric bilinear form on ${\displaystyle \mathbb {R} ^{3}}$  is a wedge product of two covectors.