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

Linear functions preserving Green's relations over fields