In mathematics, the complex conjugate of a complexvector space is a complex vector space that 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 (different multiplication by ).
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 the 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 that 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-dagger, a row vector that 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
Antidual space – Conjugate homogeneous additive mapPages displaying short descriptions of redirect targets
^K. Schmüdgen (11 November 2013). Unbounded Operator Algebras and Representation Theory. Birkhäuser. p. 16. ISBN978-3-0348-7469-4.
Further readingedit
Budinich, P. and Trautman, A. The Spinorial Chessboard. Springer-Verlag, 1988. ISBN0-387-19078-3. (complex conjugate vector spaces are discussed in section 3.3, pag. 26).
May 29, 2024
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In mathematics the complex conjugate of a complex vector space V displaystyle V is a complex vector space V displaystyle overline V that has the same elements and additive group structure as V displaystyle V but whose scalar multiplication involves conjugation of the scalars In other words the scalar multiplication of V displaystyle overline V satisfiesa v a v displaystyle alpha v overline alpha cdot v where displaystyle is the scalar multiplication of V displaystyle overline V and displaystyle cdot is the scalar multiplication of V displaystyle V The letter v displaystyle v stands for a vector in V displaystyle V a displaystyle alpha is a complex number and a displaystyle overline alpha denotes the complex conjugate of a displaystyle alpha 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 displaystyle J different multiplication by i displaystyle i Contents 1 Motivation 2 Complex conjugation functor 3 Structure of the conjugate 4 Complex conjugate of a Hilbert space 5 See also 6 References 7 Further readingMotivation editIf V displaystyle V nbsp and W displaystyle W nbsp are complex vector spaces a function f V W displaystyle f V to W nbsp is antilinear iff v w f v f w and f a v a f v displaystyle f v w f v f w quad text and quad f alpha v overline alpha f v nbsp With the use of the conjugate vector space V displaystyle overline V nbsp an antilinear map f V W displaystyle f V to W nbsp can be regarded as an ordinary linear map of type V W displaystyle overline V to W nbsp The linearity is checked by noting f a v f a v a f v a f v displaystyle f alpha v f overline alpha cdot v overline overline alpha cdot f v alpha cdot f v nbsp Conversely any linear map defined on V displaystyle overline V nbsp gives rise to an antilinear map on V displaystyle V nbsp This is the same underlying principle as in defining the opposite ring so that a right R displaystyle R nbsp module can be regarded as a left R o p displaystyle R op nbsp module or that of an opposite category so that a contravariant functor C D displaystyle C to D nbsp can be regarded as an ordinary functor of type C o p D displaystyle C op to D nbsp Complex conjugation functor editA linear map f V W displaystyle f V to W nbsp gives rise to a corresponding linear map f V W displaystyle overline f overline V to overline W nbsp that has the same action as f displaystyle f nbsp Note that f displaystyle overline f nbsp preserves scalar multiplication becausef a v f a v a f v a f v displaystyle overline f alpha v f overline alpha cdot v overline alpha cdot f v alpha overline f v nbsp Thus complex conjugation V V displaystyle V mapsto overline V nbsp and f f displaystyle f mapsto overline f nbsp define a functor from the category of complex vector spaces to itself If V displaystyle V nbsp and W displaystyle W nbsp are finite dimensional and the map f displaystyle f nbsp is described by the complex matrix A displaystyle A nbsp with respect to the bases B displaystyle mathcal B nbsp of V displaystyle V nbsp and C displaystyle mathcal C nbsp of W displaystyle W nbsp then the map f displaystyle overline f nbsp is described by the complex conjugate of A displaystyle A nbsp with respect to the bases B displaystyle overline mathcal B nbsp of V displaystyle overline V nbsp and C displaystyle overline mathcal C nbsp of W displaystyle overline W nbsp Structure of the conjugate editThe vector spaces V displaystyle V nbsp and V displaystyle overline V nbsp have the same dimension over the complex numbers and are therefore isomorphic as complex vector spaces However there is no natural isomorphism from V displaystyle V nbsp to V displaystyle overline V nbsp The double conjugate V displaystyle overline overline V nbsp is identical to V displaystyle V nbsp Complex conjugate of a Hilbert space editGiven a Hilbert space H displaystyle mathcal H nbsp either finite or infinite dimensional its complex conjugate H displaystyle overline mathcal H nbsp is the same vector space as its continuous dual space H displaystyle mathcal H prime nbsp There is one to one antilinear correspondence between continuous linear functionals and vectors In other words any continuous linear functional on H displaystyle mathcal H nbsp is an inner multiplication to some fixed vector and vice versa citation needed Thus the complex conjugate to a vector v displaystyle v nbsp particularly in finite dimension case may be denoted as v displaystyle v dagger nbsp v dagger a row vector that is the conjugate transpose to a column vector v displaystyle v nbsp In quantum mechanics the conjugate to a ket vector ps displaystyle psi rangle nbsp is denoted as ps displaystyle langle psi nbsp a bra vector see bra ket notation See also editAntidual space Conjugate homogeneous additive mapPages displaying short descriptions of redirect targets Linear complex structure Mathematics concept Riesz representation theorem Theorem about the dual of a Hilbert space conjugate bundleReferences edit K Schmudgen 11 November 2013 Unbounded Operator Algebras and Representation Theory Birkhauser p 16 ISBN 978 3 0348 7469 4 Further reading editBudinich 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 Retrieved from https en wikipedia org w index php title Complex conjugate of a vector space amp oldid 1189557167, wikipedia, wiki, book, books, library,