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In [[mathematics]], a [[map (mathematics)|mapping]] <math>f:V\to W</math> from a [[complex vector space]] to another is said to be '''antilinear''' (or '''conjugate-linear''' or '''[[semilinear transformation|semilinear]]''', though the latter term is more general) if


:<math>f(ax+by)=\bar{a}f(x)+\bar{b}f(y)</math>


for all ''a'', ''b'' in '''C''' and all ''x'', ''y'' in ''V'', where <math>\bar{a}</math> and <math>\bar{b}</math> are the [[complex conjugate]]s of ''a'' and ''b'' respectively. The [[composition (mathematics)|composition]] of two antilinear maps is complex-[[linear]].
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An antilinear map <math>f:V\to W</math> may be equivalently described in terms of the [[linear map]] <math>\bar f:V\to\bar W</math> from <math>V</math> to the [[complex conjugate vector space]] <math>\bar W</math>.
 
Antilinear maps occur in quantum mechanics in the study of [[T-symmetry|time reversal]] and in [[spinor calculus]], where it is customary to replace the bars over the basis vectors and the components of geometric objects by dots put above the indices.  
 
==References==
* Horn and Johnson, ''Matrix Analysis,'' Cambridge University Press, 1985. ISBN 0-521-38632-2. (antilinear maps are discussed in section 4.6).
* Budinich, P. and Trautman, A. ''The Spinorial Chessboard''. Spinger-Verlag, 1988. ISBN 0-387-19078-3. (antilinear maps are discussed in section 3.3).
 
== See also ==
*[[Linear map]]
*[[Complex conjugate]]
*[[Sesquilinear form]]
*[[Matrix consimilarity]]
*[[T-symmetry|Time reversal]]
 
[[Category:Functions and mappings]]
[[Category:Linear algebra]]

Latest revision as of 00:06, 12 December 2014


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