Meanings
- not comparablemathematicsHaving 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.
1 more example
(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 - Whose sign changes on the application of a matrix transpose or some generalisation thereof:
- not comparablelinear algebraWhose 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 - not comparablelinear algebraThat 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 - not comparablelinear algebraFor 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
Synonyms & antonyms
Origin
- Proto-Indo-European*h₂ent-
- Proto-Indo-European*h₂énts
- Proto-Indo-European*h₂énti
- Ancient Greekᾰ̓ντῐ́ (ăntĭ́)
- Ancient Greekἀντι- (anti-)
- Englishanti-
- Englishsymmetric
- Englishantisymmetric
Translations
Czech antisymetrický
- antisymetrický(order theory; of a binary relation on a set)
Esperanto malsimetria, antisimetria
- malsimetria(order theory; of a binary relation on a set)
- antisimetria(order theory; of a binary relation on a set)
Finnish antisymmetrinen, antisymmetrinen
- antisymmetrinen(order theory; of a binary relation on a set)
- antisymmetrinen(linear algebra)
French antisymétrique
- antisymétrique(order theory; of a binary relation on a set)
German antisymmetrisch, antisymmetrisch
- antisymmetrisch(order theory; of a binary relation on a set)
- antisymmetrisch(linear algebra)
Greek αντισυμμετρικός, αντισυμμετρικός
- αντισυμμετρικός (antisymmetrikós)(order theory; of a binary relation on a set)
- αντισυμμετρικός (antisymmetrikós)(linear algebra)
Icelandic andsamhverfur
- andsamhverfur(order theory; of a binary relation on a set)
Japanese 反対称的
- 反対称的 (han-taishō-teki)(order theory; of a binary relation on a set)
Polish antysymetryczny, antysymetryczny
- antysymetryczny(order theory; of a binary relation on a set)
- antysymetryczny(linear algebra)
Portuguese antissimétrico
- antissimétrico(order theory; of a binary relation on a set)
Romanian antisimetric, antisimetric
- antisimetric(order theory; of a binary relation on a set)
- antisimetric(linear algebra)
Russian антисимметри́чный
- антисимметри́чный (antisimmetríčnyj)(order theory; of a binary relation on a set)
Spanish antisimétrico, antisimétrico
- antisimétrico(order theory; of a binary relation on a set)
- antisimétrico(linear algebra)
Swedish antisymmetrisk
- antisymmetrisk(order theory; of a binary relation on a set)
Cite this entry
antisymmetric. (n.d.). In WatkinsX Dictionary. Retrieved November 4, 2026, from https://dictionary.watkinsx.com/word/antisymmetric
“antisymmetric.” WatkinsX Dictionary, https://dictionary.watkinsx.com/word/antisymmetric. Accessed November 4, 2026.
WatkinsX Dictionary, s.v. “antisymmetric,” accessed November 4, 2026, https://dictionary.watkinsx.com/word/antisymmetric.
Source: antisymmetric on Wiktionary, CC BY-SA 4.0.