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In mathematics, the complex conjugate of a
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
V\, is a complex vector space \overline V, which has the same elements and additive group structure as V, but whose
scalar multiplication In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector b ...
involves conjugation of the scalars. In other words, the scalar multiplication of \overline V satisfies \alpha\,*\, v = where * is the scalar multiplication of \overline and \cdot is the scalar multiplication of V. The letter v stands for a vector in V, \alpha is a complex number, and \overline denotes the
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
of \alpha. More concretely, the complex conjugate vector space is the same underlying vector space (same set of points, same vector addition and real scalar multiplication) with the conjugate
linear complex structure In mathematics, a complex structure on a real vector space ''V'' is an automorphism of ''V'' that squares to the minus identity, −''I''. Such a structure on ''V'' allows one to define multiplication by complex scalars in a canonical fashion so ...
J (different multiplication by i).


Motivation

If V and W are complex vector spaces, a function f : V \to W is
antilinear In mathematics, a function f : V \to W between two complex vector spaces is said to be antilinear or conjugate-linear if \begin f(x + y) &= f(x) + f(y) && \qquad \text \\ f(s x) &= \overline f(x) && \qquad \text \\ \end hold for all vectors x, y ...
if f(v + w) = f(v) + f(w) \quad \text \quad f(\alpha v) = \overline \, f(v) With the use of the conjugate vector space \overline V, an antilinear map f : V \to W can be regarded as an ordinary
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
of type \overline \to W. The linearity is checked by noting: f(\alpha * v) = f(\overline \cdot v) = \overline \cdot f(v) = \alpha \cdot f(v) Conversely, any linear map defined on \overline gives rise to an antilinear map on V. This is the same underlying principle as in defining
opposite ring In mathematics, specifically abstract algebra, the opposite of a ring is another ring with the same elements and addition operation, but with the multiplication performed in the reverse order. More explicitly, the opposite of a ring is the ring w ...
so that a right R-
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Mo ...
can be regarded as a left R^-module, or that of an
opposite category In category theory, a branch of mathematics, the opposite category or dual category ''C''op of a given category ''C'' is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal twice yields t ...
so that a
contravariant functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ...
C \to D can be regarded as an ordinary functor of type C^ \to D.


Complex conjugation functor

A linear map f : V \to W\, gives rise to a corresponding linear map \overline : \overline \to \overline which has the same action as f. Note that \overline f preserves scalar multiplication because \overline(\alpha * v) = f(\overline \cdot v) = \overline \cdot f(v) = \alpha * \overline(v) Thus, complex conjugation V \mapsto \overline and f \mapsto\overline f define a
functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
from the
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...
of complex vector spaces to itself. If V and W are finite-dimensional and the map f is described by the complex
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
A with respect to the bases \mathcal of V and \mathcal of W, then the map \overline is described by the complex conjugate of A with respect to the bases \overline of \overline and \overline of \overline.


Structure of the conjugate

The vector spaces V and \overline have the same
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
over the complex numbers and are therefore isomorphic as complex vector spaces. However, there is no
natural isomorphism In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
from V to \overline. The double conjugate \overline is identical to V.


Complex conjugate of a Hilbert space

Given a Hilbert space \mathcal (either finite or infinite dimensional), its complex conjugate \overline is the same vector space as its
continuous dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
\mathcal^. There is one-to-one antilinear correspondence between continuous linear functionals and vectors. In other words, any continuous
linear functional In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , the ...
on \mathcal is an inner multiplication to some fixed vector, and vice versa. Thus, the complex conjugate to a vector v, particularly in finite dimension case, may be denoted as v^\dagger (v-dagger, a
row vector In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example, \boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end. Similarly, a row vector is a 1 \times n matrix for some n, c ...
which is the
conjugate transpose In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex c ...
to a column vector v). In quantum mechanics, the conjugate to a ''ket vector'' \,, \psi\rangle is denoted as \langle\psi, \, – a ''bra vector'' (see bra–ket notation).


See also

* * * *
conjugate bundle In mathematics, a complex vector bundle is a vector bundle whose fibers are complex vector spaces. Any complex vector bundle can be viewed as a real vector bundle through the restriction of scalars. Conversely, any real vector bundle ''E'' can be ...


References


Further reading

* Budinich, P. and Trautman, A. ''The Spinorial Chessboard''. Springer-Verlag, 1988. {{ISBN, 0-387-19078-3. (complex conjugate vector spaces are discussed in section 3.3, pag. 26). Linear algebra
Vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...