Linear/additive preservers of rank 2 on spaces of alternate matrices over fields
A matrix \(A\in M_n(F)\) is said to be alternate if \(x^tAx=0\) for every \(x\in M_{n,1}(F)\). If the characteristic of \(F\) is not 2, \(A\) is alternate iff \(A\) is skew-symmetric, and if \(\text{char}(F)=2\), \(A\) is alternate iff it is symmetric. The author obtains a characterization of additive transformations on the space of alternate matrices which preserve the set of rank-2 matrices on the fields which are not isomorphic to their proper subfields. The characterization of the corresponding linear transformations is given for an arbitrary field.
scientific article; zbMATH DE number 2136673