Linear spaces and preservers of symmetric matrices of bounded rank-two

摘要

Let F{double-struck} be a field of characteristic two. Let S_n(F{double-struck}) denote the vector space of all n × n symmetric matrices over F{double-struck}. We characterize (i) subspaces of S_n(F{double-struck}) all whose elements have rank at most two where n ≥ 3, (ii) linear maps from S_m(F{double-struck}) to S_n(F{double-struck}) that sends matrices of rank at most two into matrices of rank at most two where m, n ≥ 3 and {pipe}F{double-struck}{pipe} ≠ 2.