Staggered Array Of Security Rank Equivalent To The Addition Mapping

Linear preserver problems are an active research area in matrix theory. Itmainly characterizes the linear and additive operators preserving invariant(funtion,subset, relation). There are some results preserving certain equivalence relationson matrix spaces. For example, Li e.t.al, studied nonzero linear operator that pre-serves rank equivalence under the condition of bijectivity on M_m×n(C), where Cis the field of complex numbers, Horn e.t.al, studied nonzero linear operator thatpreserves rank equivalence without the condition of bijectivity on M_m×n(C). It isinteresting for many researchers to study the linear and additive preserver problemson different matrices. Alternate matrices have close relation with quadratic formsand symplectic group of classical groups, meanwhile, orthogonal Lie algebra of lin-ear Lie algebra consists of alternate matrices. So it is worthwhile and interestingto study the preserving problems of alternate matrices. Let F be any field, F^* beits subset consisting of all nonzero elements, and m, n≥4 be any integer. ForA∈K_n(F), we denote by A^t the transpose of A, denote by rank(A) the rank of A.A square matrix A is said to be alternate if A^t=-A and all diagonal elements arezeros. Denote by K_n(F) the space of all n×n alternate matrices over F. Recently,Tang Xiaomin, Chen Xuemei, studied nonzero linear operator that preserved rankequivalence from K_n(F) to K_n(F). In this paper, we extend the above referenceresult to addition, we use alternate matrix geometric fundamental theorem to char-acterize the additive operator preserving rank equivalence from K_n(F) to K_n(F),some applications include severals aspects as following:(1) characterizing the additive operator preserving rank from K_n(F) to K_n(F),obtaining that the additive operator preserving rank is necessarily preserving rankequivalence and applying the result preserving rank equivalence to preserving rank.(2) proving that preserving rank non-increasing relation, preserving rank equiv-alence, and preserving rank are all equivalent. Furthermore, we characterize the ad-ditive operator preserving the above rank relation using the result preserving rankequivalence.