I know of Choi's theorem and some related problems, but not a solution to this exact problem:



> 
> Characterize the linear maps from the space $S\_n$ of symmetric $n \times n $ matrices to itself that preserve positive semidefiniteness.
> 
> 
> 


It looks a natural question; has a simple characterization been found? Where can I find it?



Linear maps preserving positive semidefiniteness
Somewhat surprisingly, this seems to be still open. It is a linear preserver problem, about which there is a nice overview [here](http://people.wm.edu/~cklixx/lpp.pdf). But your specific problem seems to be open, according to this [recent preprint](http://arxiv.org/pdf/1011.3739.pdf). They also say that in an [earlier paper](http://arxiv.org/pdf/1008.1347v1.pdf) they settled the problem with some extra assumptions.


