Complex conjugate of a vector space
Template:Short description In mathematics, the complex conjugate of a complex vector space <math>V\,</math> is a complex vector space <math>\overline V</math> that has the same elements and additive group structure as <math>V,</math> but whose scalar multiplication involves conjugation of the scalars. In other words, the scalar multiplication of <math>\overline V</math> satisfies <math display="block">\alpha\,*\, v = {\,\overline{\alpha} \cdot \,v\,}</math> where <math>*</math> is the scalar multiplication of <math>\overline{V}</math> and <math>\cdot</math> is the scalar multiplication of <math>V.</math> The letter <math>v</math> stands for a vector in <math>V,</math> <math>\alpha</math> is a complex number, and <math>\overline{\alpha}</math> denotes the complex conjugate of <math>\alpha.</math><ref name="Schmüdgen2013">Template:Cite book</ref>
More concretely, the complex conjugate vector space is the same underlying Template:Em vector space (same set of points, same vector addition and real scalar multiplication) with the conjugate linear complex structure <math>J</math> (different multiplication by <math>i</math>).
MotivationEdit
If <math>V</math> and <math>W</math> are complex vector spaces, a function <math>f : V \to W</math> is antilinear if <math display="block">f(v + w) = f(v) + f(w) \quad \text{ and } \quad f(\alpha v) = \overline{\alpha} \, f(v)</math> With the use of the conjugate vector space <math>\overline V</math>, an antilinear map <math>f : V \to W</math> can be regarded as an ordinary linear map of type <math>\overline{V} \to W.</math> The linearity is checked by noting: <math display="block">f(\alpha * v) = f(\overline{\alpha} \cdot v) = \overline{\overline{\alpha}} \cdot f(v) = \alpha \cdot f(v)</math> Conversely, any linear map defined on <math>\overline{V}</math> gives rise to an antilinear map on <math>V.</math>
This is the same underlying principle as in defining the opposite ring so that a right <math>R</math>-module can be regarded as a left <math>R^{op}</math>-module, or that of an opposite category so that a contravariant functor <math>C \to D</math> can be regarded as an ordinary functor of type <math>C^{op} \to D.</math>
Complex conjugation functorEdit
A linear map <math>f : V \to W\,</math> gives rise to a corresponding linear map <math>\overline{f} : \overline{V} \to \overline{W}</math> that has the same action as <math>f.</math> Note that <math>\overline f</math> preserves scalar multiplication because <math display="block">\overline{f}(\alpha * v) = f(\overline{\alpha} \cdot v) = \overline{\alpha} \cdot f(v) = \alpha * \overline{f}(v)</math> Thus, complex conjugation <math>V \mapsto \overline{V}</math> and <math>f \mapsto\overline f</math> define a functor from the category of complex vector spaces to itself.
If <math>V</math> and <math>W</math> are finite-dimensional and the map <math>f</math> is described by the complex matrix <math>A</math> with respect to the bases <math>\mathcal{B}</math> of <math>V</math> and <math>\mathcal{C}</math> of <math>W,</math> then the map <math>\overline{f}</math> is described by the complex conjugate of <math>A</math> with respect to the bases <math>\overline{\mathcal{B}}</math> of <math>\overline{V}</math> and <math>\overline{\mathcal{C}}</math> of <math>\overline{W}.</math>
Structure of the conjugateEdit
The vector spaces <math>V</math> and <math>\overline{V}</math> have the same dimension over the complex numbers and are therefore isomorphic as complex vector spaces. However, there is no natural isomorphism from <math>V</math> to <math>\overline{V}.</math>
The double conjugate <math>\overline{\overline{V}}</math> is identical to <math>V.</math>
Complex conjugate of a Hilbert spaceEdit
Given a Hilbert space <math>\mathcal{H}</math> (either finite or infinite dimensional), its complex conjugate <math>\overline{\mathcal{H}}</math> is the same vector space as its continuous dual space <math>\mathcal{H}^{\prime}.</math> There is one-to-one antilinear correspondence between continuous linear functionals and vectors. In other words, any continuous linear functional on <math>\mathcal{H}</math> is an inner multiplication to some fixed vector, and vice versa.Template:Citation needed
Thus, the complex conjugate to a vector <math>v,</math> particularly in finite dimension case, may be denoted as <math>v^\dagger</math> (v-dagger, a row vector that is the conjugate transpose to a column vector <math>v</math>). In quantum mechanics, the conjugate to a ket vector <math>\,|\psi\rangle</math> is denoted as <math>\langle\psi|\,</math> – a bra vector (see bra–ket notation).
See alsoEdit
ReferencesEdit
Further readingEdit
- Budinich, P. and Trautman, A. The Spinorial Chessboard. Springer-Verlag, 1988. Template:ISBN. (complex conjugate vector spaces are discussed in section 3.3, pag. 26).