antisymmetric

== English == === Etymology === From anti- +‎ symmetric. === Pronunciation === === Adjective === antisymmetric (not comparable) (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, 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. (linear algebra, of certain mathematical objects) Whose sign changes on the application of a matrix transpose or some generalisation thereof: (of a matrix) Whose transpose equals its negative (i.e., MT = −M); (of a tensor) That changes sign when any two indices are interchanged (e.g., Tijk = -Tjik); (of a bilinear form) For which B(w,v) = -B(v,w). ==== Synonyms ==== (linear algebra): skew-symmetric ==== Derived terms ==== ==== Related terms ==== antisymmetry symmetric ==== Translations ==== === See also === anticommutative skew-symmetric === Further reading === Antisymmetric relation on Wikipedia.Wikipedia Asymmetric relation on Wikipedia.Wikipedia Symmetric relation on Wikipedia.Wikipedia Antisymmetric matrix on Wikipedia.Wikipedia Antisymmetric tensor on Wikipedia.Wikipedia