  We study linear functions on the space of $n \times n$ matrices over a field
which preserve or strongly preserve each of Green's equivalence relations
($\mathcal{L}$, $\mathcal{R}$, $\mathcal{H}$ and $\mathcal{J}$) and the
corresponding pre-orders. For each of these relations we are able to completely
describe all preservers over an algebraically closed field (or more generally,
a field in which every polynomial of degree $n$ has a root), and all strong
preservers and bijective preservers over any field. Over a general field, the
non-zero $\mathcal{J}$-preservers are all bijective and coincide with the
bijective rank-$1$ preservers, while the non-zero $\mathcal{H}$-preservers turn
out to be exactly the invertibility preservers, which are known. The
$\mathcal{L}$- and $\mathcal{R}$-preservers over a field with "few roots" seem
harder to describe: we give a family of examples showing that they can be quite
wild.
