Staggered Matrix Space Remain Problems

Suppose R is a field of real numbers and F is a field with chF = 2. Let n≥4be an integer, K_n (R) and K_n (F) denote the set of all n×n antisymmetric matricesover R and the set of n×n alternate matrices over F, respectively. Denote byωK_n (F) the subset of K_n (F) consisting of all max-rank matrices. If A3 = ?A,A∈Rn×n, A is called an antitripotence matrix. A matrix A∈K_n (F) is saidto have max-rank if rank(A) = n (resp., n ? 1 ) when n is even(resp.,odd). Anadditive mapφ: K_n (F)â†'K_n (F) is said to preserve the max-rank ifφ(ωK_n (F)) =ωK_n (F). An additive mapφ: K_n (F)â†'K_n (F) is said to preserve the determinantif detφA = detA. In chapter 2, this paper characterizes linear operators preservingantitripotence on K_n (R) by using the result about the linear rank-2 preserverson K_n (F); In chapter 3, we describe the additive max-rank preservers on K_n (F)by using the result about the additive rank-2 preservers on K_n (F). Furthermore,we, applying the result about additive preservers of the max-rank, characterize theadditive preservers of the determinant on K_n (F).