antisymmetric

adj

1. (not-comparable)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.”

   “(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.”

2. (not-comparable)Whose sign changes on the application of a matrix transpose or some generalisation thereof:

   “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.”

3. (not-comparable)Whose sign changes on the application of a matrix transpose or some generalisation thereof:

   “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.”

4. (not-comparable)Whose sign changes on the application of a matrix transpose or some generalisation thereof:

   “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.”