different between symmetric vs antisymmetric

English

Etymology

From Latin symmetria from Ancient Greek ????????? (summetría).

Pronunciation

• IPA^(key): /s??m?t??k/

Adjective

symmetric (comparative more symmetric, superlative most symmetric)

1. Symmetrical.
2. (set theory) Of a relation R on a set S, such that xRy if and only if yRx for all members x and y of S (that is, if the relation holds between any element and a second, it also holds between the second and the first).

   "Is a sibling of" is a symmetric relation.

3. (cryptography) Using the same key (or keys that are trivially related) for both encryption and decryption.

Related terms

• asymmetric
• symmetry

Translations

See also

• reflexive
• transitive

Anagrams

• mycterism

English

Etymology

anti- +? symmetric.

Adjective

antisymmetric (not comparable)

1. (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.

   • 2006, S. C. Sharma, Metric Space, Discovery Publishing House, page 73,

     (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 ${\displaystyle \Leftrightarrow }$ 'x divides y' for all x, y ? N.

     This relation is an antisymmetric relation on N.

2. (linear algebra, of certain mathematical objects) Whose sign changes on the application of a matrix transpose or some generalisation thereof:
   1. (of a matrix) Whose transpose equals its negative (i.e., M^T = ?M);
      • 1974, Robert McCredie May, Stability and Complexity in Model Ecosystems, Princeton University Press, page 193,

        The eigenvalues of an antisymmetric matrix are all purely imaginary numbers, and occur as conjugate pairs, ${\displaystyle +iw}$ and ${\displaystyle -iw}$. 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.

   2. (of a tensor) That changes sign when any two indices are interchanged (e.g., T_ijk = -T_jik);
      • 1986, Millard F. Beatty Jr., Principles of Engineering Mechanics, Volume 1: Kinematics — The Geometry of Motion, Plenum Press, page 163,

        Notice that the tensors defined by:

            ${\displaystyle \textstyle T_{S}\equiv {\frac {1}{2}}(T+T^{T})}$,     ${\displaystyle \textstyle T_{A}\equiv {\frac {1}{2}}(T-T^{T}),}$     (3.47)

        are the symmetric and antisymmetric parts, respectively; they are known as the symmetric and antisymmetric parts of T.

   3. (of a bilinear form) For which B(w,v) = -B(v,w).
      • 2012, Stephanie Frank Singer, Symmetry in Mechanics: A Gentle, Modern Introduction, Springer, page 28,

        Antisymmetric bilinear forms and wedge products are defined exactly as above, only now they are functions from ${\displaystyle \mathbb {R} ^{n}\times \mathbb {R} ^{n}}$ to ${\displaystyle \mathbb {R} }$. […]

        Exercise 21 Show that every antisymmetric bilinear form on ${\displaystyle \mathbb {R} ^{3}}$ is a wedge product of two covectors.

Synonyms

• (linear algebra): skew-symmetric

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