Antilinear map

In mathematics, a mapping f : V → W from a complex vector space to another is said to be antilinear (or conjugate-linear or semilinear, though the latter term is more general) if

$f(ax+by)={\bar {a}}f(x)+{\bar {b}}f(y)$

for all a, b in C and all x, y in V, where ${\bar {a}}$ and ${\bar {b}}$ are the complex conjugates of a and b respectively. The composition of two antilinear maps is complex-linear.

An antilinear map $f:V\to W$ may be equivalently described in terms of the linear map ${\bar {f}}:V\to {\bar {W}}$ to the complex conjugate vector space ${\bar {W}}$.

Antilinear maps occur in quantum mechanics in the study of time reversal.

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
• consimilar matrix
• Time reversal