T_n (f) Preserves Rank Export Mapping

preserver problem include linear preserver problems, additive preserver problemsand multiplicative preserver problems. preserver problems have been studied extensively, and many interesting results have been discovered and obtained . especiallyin the most recent, the issue has become a very active research topic. On the one hand, the theoretical value become of its.On the other hand, because many of the problems in differential equations, system control, the fields of mathematical statistical has a broad background in practical applications.An extensive research on preserver problems is to characterize maps between sets preserving certain properties, functions, subsets or relations invariant.Suppose F is an arbitrary field. Let F^* be the nonzero elements of F, and let m,n be integers with min{m.n}≥2.Denote by M_mn(F) the set of all m×n matrices over F, and by T_n(F) the set of all n×n upper triangular matrices. Let f_ij(i = 1,2,..., m, j = 1, 2,…, n) be maps from F to itself. If a map f: M_mn(F)â†'M_mn(F) be defined byThen we say that f is produced by f_ij. Furthermore, we say that f preserve rank -1 matrices if rankf(A) = 1. for every rank - 1 matrices A∈M_mn(F).In this paper.At first, in Chapter 2 section 1, we describe the structure of all f is produced by f_ij preservers rank-1 from T_n(F)to T_n(F), and thereby, in Chapter 2 section 2 the general form of all f is produced by f_ij preservers rank from T_n(F)to T_n(F) is determined as an application.