Antilinear map
Encyclopedia
In mathematics
, a mapping
from a complex vector space
to another is said to be antilinear (or conjugate-linear or semilinear, though the latter term is more general) if
for all a, b in C and all x, y in V, where
and 
are the complex conjugate
s of a and b respectively. The composition of two antilinear maps is complex-linear
.
An antilinear map
may be equivalently described in terms of the linear map 
from 
to the complex conjugate vector space
.
Antilinear maps occur in quantum mechanics in the study of time reversal
and in spinor calculus, where it is customary to replace the bars over the basis vectors and the components of geometric objects by dots put above the indices.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a mapping
Map (mathematics)
In most of mathematics and in some related technical fields, the term mapping, usually shortened to map, is either a synonym for function, or denotes a particular kind of function which is important in that branch, or denotes something conceptually similar to a function.In graph theory, a map is a...
from a complex vector spaceComplex vector space
A complex vector space is a vector space over the complex numbers. It can also refer to:* a vector space over the real numbers with a linear complex structure...
to another is said to be antilinear (or conjugate-linear or semilinear, though the latter term is more general) if
for all a, b in C and all x, y in V, where
and are the complex conjugateComplex conjugate
In mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...
s of a and b respectively. The composition of two antilinear maps is complex-linear
Linear
In mathematics, a linear map or function f is a function which satisfies the following two properties:* Additivity : f = f + f...
.
An antilinear map
may be equivalently described in terms of the linear map from to the complex conjugate vector spaceComplex conjugate vector space
In mathematics, the complex conjugate of a complex vector space V\, is the complex vector space \overline V consisting of all formal complex conjugates of elements of V\,...
.Antilinear maps occur in quantum mechanics in the study of time reversal
Time reversal
Time reversal may refer to:* In physics, T-symmetry - the study of thermodynamics and the symmetry of certain physical laws where the concept of time is reversed — ie...
and in spinor calculus, where it is customary to replace the bars over the basis vectors and the components of geometric objects by dots put above the indices.
See also
- Linear map
- Complex conjugateComplex conjugateIn mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...
- Sesquilinear formSesquilinear formIn mathematics, a sesquilinear form on a complex vector space V is a map V × V → C that is linear in one argument and antilinear in the other. The name originates from the numerical prefix sesqui- meaning "one and a half"...
- consimilar matrix
- Time reversalTime reversalTime reversal may refer to:* In physics, T-symmetry - the study of thermodynamics and the symmetry of certain physical laws where the concept of time is reversed — ie...