# Linear maps preserving positive semidefiniteness

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?

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What's Choi's theorem? – Felix Goldberg Jan 3 '13 at 14:45
@FelixGoldberg en.wikipedia.org/wiki/… In short, it says that all completely positive linear maps (a stronger condition than preserving positive semidefiniteness) are precisely those in the form $\Phi(X)=\sum W_i X W_i^*$ – Federico Poloni Jan 3 '13 at 15:19