5.71.212.84: /* Models */ used prime symbol to avoid mistaken interpretation of apostrophes as markup

2014-10-23T09:46:24Z

Models: used prime symbol to avoid mistaken interpretation of apostrophes as markup

← Older revision		Revision as of 11:46, 23 October 2014
Line 1:		Line 1:
	In [[mathematics]], the '''(formal) complex conjugate''' of a [[complex numbers\|complex]] [[vector space]] <math>V\,</math> is the complex vector space <math>\overline V</math> consisting of all formal [[complex conjugate]]s of elements of <math>V\,</math>. That is, <math>\overline V</math> is a vector space whose elements are in [[bijection\|one-to-one correspondence]] with the elements of <math>V\,</math>:		Plastic and Reconstructive Surgeon Rolando from Blenheim, has interests including garage saleing, new launch property singapore and snorkeling. Finished a cruise ship experience that included passing by Vallée de Mai Nature Reserve.<br><br>Feel free to visit my blog post :: [http://gamerhop.com/groups/singapore-rental/ gamerhop.com]
	~~:<math>\overline V = \{\overline v \mid v \in V\},</math>~~
	~~with the following rules for [[addition]]~~ and ~~[[scalar multiplication]]:~~
	~~:<math>\overline v + \overline w = \overline{\~~,~~v+w\~~,~~}\quad\text{~~and~~}\quad\alpha\,\overline v = \overline{\,\overline \alpha \,v\,}~~.~~</math>~~
	~~Here <math>v\,</math> and <math>w\,</math> are vectors in <math>V\,</math>, <math>\alpha\,</math> is~~ a ~~complex number, and <math>\overline\alpha</math> denotes the complex conjugate of <math>\alpha\,</math>.~~

	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'').

	~~==Antilinear maps==~~
	~~If <math>V\,</math> and <math>W\,</math> are complex vector spaces, a function <math>f\colon V \to W\,</math> is [[antilinear]] if~~
	~~:<math>f(v+v') = f(v) + f(v')\quad\text{and}\quad f(\alpha v) = \overline\alpha \, f(v)</math>~~
	~~for all <math>v,v'\in V\,</math> and <math>\alpha\in\mathbb{C}</math>.~~

	~~One reason to consider the vector space <math>\overline V</math> is~~ that ~~it makes antilinear maps into [[linear map]]s. Specifically, if <math>f\colon V \to W\,</math> is an antilinear map, then the corresponding map <math>\overline V \to W</math> defined~~ by
	~~:<math>\overline v \mapsto f(v)</math>~~
	~~is linear~~. ~~Conversely, any linear map defined on~~ <~~math~~>~~\overline V~~<~~/math~~> ~~gives rise~~ to ~~an antilinear map on <math>V\,</math>.~~

	~~One way of thinking about this correspondence is that the map <math>C\colon V \to \overline V</math> defined by~~
	:~~<math>C(v) = \overline v</math>~~
	is an antilinear bijection. Thus if <math>f\colon \overline V \to W</math> is linear, then [[Function composition\|composition]] <math>f \circ C\colon V \to W\,</math> is antilinear, and ''vice versa''.

	~~==Conjugate linear maps==~~
	~~Any linear map <math>f \colon V \to W\,</math> induces a '''conjugate linear map''' <math>\overline f \colon \overline V \to \overline W</math>, defined by the formula~~
	:~~<math>\overline f (\overline v) = \overline{\,f(v)\,}.</math>~~
	~~The conjugate linear map <math>\overline f</math> is linear. Moreover, the~~ [~~[identity function\|identity map]] on <math>V\,</math> induces the identity map <math>\overline V</math>, and~~
	:~~<math>\overline f \circ \overline g = \overline{\,f \circ g\,}<~~/~~math>~~
	~~for any two linear maps <math>f\,</math> and <math>g\,<~~/~~math>~~. ~~Therefore, the rules <math>V\mapsto \overline V<~~/~~math> and <math>f\mapsto\overline f</math> define a [[functor]] from the [[category theory\|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 (mathematics)\|matrix]] <math>A\,</math> with respect to the [[basis of a vector space\|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 conjugate==~~
	The vector spaces <math>V\,</math> and <math>\overline V</math> have the same [[dimension of a vector space\|dimension]] over the complex numbers and are therefore [[isomorphism\|isomorphic]] as complex vector spaces. However, there is no [[natural isomorphism]] from <math>V\,</math> to <math>\overline V</math>. (The map <math>C\,</math> is not an isomorphism, since it is antilinear.)

	~~The double conjugate <math>\overline{\overline V}</math> is naturally isomorphic to <math>V\,</math>, with the isomorphism <math>\overline{\overline V} \to V</math> defined by~~
	~~:<math>\overline{\overline v} \mapsto v.</math>~~
	~~Usually the double conjugate of <math>V\,</math> is simply identified with <math>V\,</math>.~~

	~~== Complex conjugate of a Hilbert space ==~~
	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}'</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.~~

	Thus, the complex conjugate to a vector <math>v</math>, particularly in finite dimension case, may be denoted as <math>v^*</math> (v-star, a [[row vector]] which 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 also==~~
	* [[Linear complex structure]]

	~~==References==~~
	* Budinich, P. and Trautman, A. ''The Spinorial Chessboard''. Spinger-Verlag, 1988. ISBN 0-387-19078-3. (complex conjugate vector spaces are discussed in section 3.3, pag. 26).



	~~[[Category:Linear algebra]]~~
	~~[[Category:Vectors\|Vector space]~~]

160.45.152.6: /* See also */ : Adding Metatheory, Object language.

2013-05-21T12:07:26Z

See also: : Adding Metatheory, Object language.

Show changes

en>ClueBot NG: Reverting possible vandalism by MassHysteriaHD to version by AnomieBOT. False positive? Report it. Thanks, ClueBot NG. (1180036) (Bot)

2012-08-23T03:50:34Z

Reverting possible vandalism by MassHysteriaHD to version by AnomieBOT. False positive? Report it. Thanks, ClueBot NG. (1180036) (Bot)

Show changes

Object theory - Revision history

5.71.212.84: /* Models */ used prime symbol to avoid mistaken interpretation of apostrophes as markup

160.45.152.6: /* See also */ : Adding Metatheory, Object language.

en>ClueBot NG: Reverting possible vandalism by MassHysteriaHD to version by AnomieBOT. False positive? Report it. Thanks, ClueBot NG. (1180036) (Bot)