To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Added in 24 Hours
Languages
Recent
Show all languages
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds

Complex conjugate of a vector space

From Wikipedia, the free encyclopedia

In mathematics, the complex conjugate of a complex vector 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 ).

Motivation

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 functor

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 conjugate

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 space

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 also

References

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

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).
This page was last edited on 12 December 2023, at 16:01
Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.