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Words, grammatical forms and meanings linked to the ontology.

antisymmetric en · ADJ

Pronunciation

Etymology

Etymology tree Proto-Indo-European *h₂ent- Proto-Indo-European *-s Proto-Indo-European *h₂énts Proto-Indo-European *-i Proto-Indo-European *h₂énti Ancient Greek ᾰ̓ντῐ́ (ăntĭ́) Ancient Greek ἀντι- (anti-)der. English anti- English symmetric English antisymmetric From anti- + symmetric.

Meanings

  1. (not-comparable, 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.
    • (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:
    • 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.
  2. (not-comparable, of a matrix, of certain mathematical objects) Whose sign changes on the application of a matrix transpose or some generalisation thereof: ▸ 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. (not-comparable, of a tensor, of certain mathematical objects) Whose sign changes on the application of a matrix transpose or some generalisation thereof: ▸ 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. (not-comparable, of a bilinear form, of certain mathematical objects) Whose sign changes on the application of a matrix transpose or some generalisation thereof: ▸ 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:

Deriveds

antisymmetrically · antisymmetricity

Relateds

anticommutative · symmetric · antisymmetry · skew-symmetric

Synonyms

skew-symmetric

Translations (21)

es antisimétrico ((linear algebra)) · eo antisimetria ((order theory; of a binary relation on a set)) · ru антисимметри́чный ((order theory; of a binary relation on a set)) · fr antisymétrique ((order theory; of a binary relation on a set)) · es antisimétrico ((order theory; of a binary relation on a set)) · de antisymmetrisch ((linear algebra)) · pl antysymetryczny ((linear algebra)) · sv antisymmetrisk ((order theory; of a binary relation on a set)) · fi antisymmetrinen ((linear algebra)) · el αντισυμμετρικός ((order theory; of a binary relation on a set)) · ro antisimetric ((order theory; of a binary relation on a set)) · el αντισυμμετρικός ((linear algebra)) · pt antissimétrico ((order theory; of a binary relation on a set)) · de antisymmetrisch ((order theory; of a binary relation on a set)) · fi antisymmetrinen ((order theory; of a binary relation on a set)) · is andsamhverfur ((order theory; of a binary relation on a set)) · pl antysymetryczny ((order theory; of a binary relation on a set)) · eo malsimetria ((order theory; of a binary relation on a set)) · cs antisymetrický ((order theory; of a binary relation on a set)) · ro antisimetric ((linear algebra)) · ja 反対称的 ((order theory; of a binary relation on a set))