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In mathematics, the complex conjugate of a complex vector space is a complex vector space , which has the same elements and additive group structure as , but whose scalar multiplication involves conjugation of the scalars. In other words, the scalar multiplication of satisfies

where is the scalar multiplication of and is the scalar multiplication of . The letter stands for a vector in , is a complex number, and denotes the complex conjugate of .[1]

More concretely, the complex conjugate vector space is the same underlying real vector space (same set of points, same vector addition and real scalar multiplication) with the conjugate linear complex structure J (different multiplication by i).

MotivationEdit

If   and   are complex vector spaces, a function   is antilinear if

 

With the use of the conjugate vector space  , an antilinear map   can be regarded as an ordinary linear map of type  . The linearity is checked by noting:

 

Conversely, any linear map defined on   gives rise to an antilinear map on  .

This is the same underlying principle as in defining opposite ring so that a right  -module can be regarded as a left  -module, or that of an opposite category so that a contravariant functor   can be regarded as an ordinary functor of type  .

Complex conjugation functorEdit

A linear map   gives rise to a corresponding linear map   which has the same action as  . Note that   preserves scalar multiplication because

 

Thus, complex conjugation   and   define a functor from the category of complex vector spaces to itself.

If   and   are finite-dimensional and the map   is described by the complex matrix   with respect to the bases   of   and   of  , then the map   is described by the complex conjugate of   with respect to the bases   of   and   of  .

Structure of the conjugateEdit

The vector spaces   and   have the same dimension over the complex numbers and are therefore isomorphic as complex vector spaces. However, there is no natural isomorphism from   to  .

The double conjugate   is identical to  .

Complex conjugate of a Hilbert spaceEdit

Given a Hilbert space   (either finite or infinite dimensional), its complex conjugate   is the same vector space as its continuous dual space  . There is one-to-one antilinear correspondence between continuous linear functionals and vectors. In other words, any continuous linear functional on   is an inner multiplication to some fixed vector, and vice versa.[citation needed]

Thus, the complex conjugate to a vector  , particularly in finite dimension case, may be denoted as   (v-star, a row vector which is the conjugate transpose to a column vector  ). In quantum mechanics, the conjugate to a ket vector   is denoted as   – a bra vector (see bra–ket notation).

See alsoEdit

ReferencesEdit

  1. ^ K. Schmüdgen (11 November 2013). Unbounded Operator Algebras and Representation Theory. Birkhäuser. p. 16. ISBN 978-3-0348-7469-4.

Further readingEdit

  • 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).