why does every projective transformation on $\Bbb RP^2$ has a fixed point?
A friend of mine stated that it is true but i can´t think of a way to prove this.
I only remember projective transformation as a bijectiv linear map and some characteristics with cross ratio.
help would be appreciated.
If you put homogeneous coordinates on RP^2, then each projective transformation is represented by multiplication by a nonsingular $3 \times 3$ real matrix $M$.
The characteristic polynomial of $M$ is a real cubic, and therefore has at least one real root, which cannot be zero or the transformation would be singular. The eigenvector corresponding to that real root sits in a line (i.e., a point of RP^2) that's invariant under the transformation. Hence there's a fixed point.