Huaʹs fundamental theorem of the geometry of matrices

Let be the space of all n×n matrices over the field , n 2. Two matrices are adjacent if rank(A−B)=1. Huaʹs fundamental theorem of the geometry of square matrices characterizes bijective maps on that preserve adjacency in both directions. In this paper we treat a long standing open problem whether the result of Hua holds true under the weaker assumption of preserving the adjacency in one direction only. We answer this question in the affirmative in the case that every nonzero homomorphism is surjective. For example, the field of real numbers has this property. In order to prove this result we have to improve Ovchinnikovʹs characterization of automorphisms of the poset of idempotent matrices.