# Antilinear map

In mathematics, a mapping ${\displaystyle f:V\to W}$ from a complex vector space to another is said to be antilinear (or conjugate-linear) if

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

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

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

Antilinear maps occur in quantum mechanics in the study of 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. Springer-Verlag, 1988. ISBN 0-387-19078-3. (antilinear maps are discussed in section 3.3).