Hermite Matrix Space Preserves Rank Export Mapping

The study of preserver problems on spaces of matrices is a very active area in the study of international matrix theory, and has a very important applied value in many fields. such as computation, statistics, equation and so on. Rank-1 preservers are core of many preserving invariants in the study of the field, so it has caused attention of a lot of scholars. Some authors have elicited produced map becauce of actual needs and have studied producing map of preserving rank-1.At present, produced map of preserving rank-1 (preserving rank) in quadrate matrices space and symmetric matrices space have gained clear results. The considerations of the present paper were inspired by it, and studied produced map of preserving rank-1(preserving rank) in Hermite matrices space. Let C bt the complex number field, f_ij(i,j∈[n]) be the map from C to itself , H_n(C) be the set of all n x n Hermite matrice over C, f be the map guided by f_ij(i,j∈[n]) in H_n(C),viz.At the end of the article , it gives the form of produced map of preserving rank-1 in H_n(C) with n≥3 and gives the form of produced map of preserving rank in H_n(C) at the bace of it. The main result of the article is that when n is integer greater than 2, and / is the map produced by (f_ij)_n in H_n(C), then the necessary and sufficient condition of f preserving rank is that exist nonzero diagonal matrix P and injective endomorphismφof C such thathere e=±1, (?)X∈H_n(C), P is conjugated matrix of P.