antisymmetric

Adjective

antisymmetric (not comparable)

1. 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.
   • The standard example for an antisymmetric relation is the relation less than or equal to on the real number system.1987, David C. Buchthal, Douglas E. Cameron, Modern Abstract Algebra, Prindle, Weber & Schmidt, page 479
   • (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 #92;Leftrightarrow 'x divides y' for all x, y ∈ N.
     This relation is an antisymmetric relation on N.2006, S. C. Sharma, Metric Space, Discovery Publishing House, page 73
2. Whose transpose equals its negative (i.e., Mᵀ = −M);
   • The eigenvalues of an antisymmetric matrix are all purely imaginary numbers, and occur as conjugate pairs, #43;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.1974, Robert McCredie May, Stability and Complexity in Model Ecosystems, Princeton University Press, page 193
3. That changes sign when any two indices are interchanged (e.g., Tᵢⱼₖ = -Tⱼᵢₖ);
   • Notice that the tensors defined by:
     #92;textstyleT#95;S#92;equiv#92;frac#123;1#125;#123;2#125;(T#43;Tᵀ), #92;textstyleT#95;A#92;equiv#92;frac#123;1#125;#123;2#125;(T-Tᵀ), (3.47)
     are the symmetric and antisymmetric parts, respectively; they are known as the symmetric and antisymmetric parts of T.1986, Millard F. Beatty Jr., Principles of Engineering Mechanics, Volume 1: Kinematics - The Geometry of Motion, Plenum Press, page 163
4. For which B(w,v) = -B(v,w).
   • Antisymmetric bilinear forms and wedge products are defined exactly as above, only now they are functions from #92;Rⁿ#92;times#92;Rⁿ to #92;R.[…]
   • Exercise 21 Show that every antisymmetric bilinear form on #92;R³ is a wedge product of two covectors.2012, Stephanie Frank Singer, Symmetry in Mechanics: A Gentle, Modern Introduction, Springer, page 28