Antisymmetric
Encyclopedia
The word antisymmetric refers to a change to an opposite quantity when another quantity is symmetrically changed. This concept is related to that of Symmetry
and Asymmetry
. The difference between these three concepts can be simply illustrated with Latin letters. The character "A" is symmetric about the vertical axis while the character "B" is not. The character "S" is antisymmetric about the vertical axis since the left side is flipped relative to the right. A character such as "H" fits the definition of both symmetric and antisymmetric. In this case the correct term is symmetric.
.
The term "antisymmetric function" is sometimes used for odd function
, although some meanings of antisymmetric are essentiality f(-y, -x) = −f(x, y). In operator notation the latter is called anticommutativity
.
In linear algebra
and theoretical physics
, the adjective antisymmetric (or skew-symmetric) is used for matrices, tensor
s, and other objects that change sign if an appropriate operation (usually the exchange of two indices, which becomes the transposition of the matrix) is performed. See:
See also Symmetry in mathematics
.
Symmetry
Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...
and Asymmetry
Asymmetry
Asymmetry is the absence of, or a violation of, symmetry.-In organisms:Due to how cells divide in organisms, asymmetry in organisms is fairly usual in at least one dimension, with biological symmetry also being common in at least one dimension....
. The difference between these three concepts can be simply illustrated with Latin letters. The character "A" is symmetric about the vertical axis while the character "B" is not. The character "S" is antisymmetric about the vertical axis since the left side is flipped relative to the right. A character such as "H" fits the definition of both symmetric and antisymmetric. In this case the correct term is symmetric.
Mathematics
In set theory, the adjective antisymmetric usually refers to an antisymmetric relationAntisymmetric relation
In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in Xor, equivalently,In mathematical notation, this is:\forall a, b \in X,\ R \and R \; \Rightarrow \; a = bor, equivalently,...
.
The term "antisymmetric function" is sometimes used for odd function
Even and odd functions
In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power series and Fourier series...
, although some meanings of antisymmetric are essentiality f(-y, -x) = −f(x, y). In operator notation the latter is called anticommutativity
Anticommutativity
In mathematics, anticommutativity is the property of an operation that swapping the position of any two arguments negates the result. Anticommutative operations are widely used in algebra, geometry, mathematical analysis and, as a consequence, in physics: they are often called antisymmetric...
.
In linear algebra
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
and theoretical physics
Theoretical physics
Theoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
, the adjective antisymmetric (or skew-symmetric) is used for matrices, tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
s, and other objects that change sign if an appropriate operation (usually the exchange of two indices, which becomes the transposition of the matrix) is performed. See:
- antisymmetric matrix (a matrix A for which AT = −A)
- antisymmetric tensor
See also Symmetry in mathematics
Symmetry in mathematics
Symmetry occurs not only in geometry, but also in other branches of mathematics. It is actually the same as invariance: the property that something does not change under a set of transformations....
.