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Linear/additive preservers of rank 2 on spaces of alternate matrices over fields. (English) Zbl 1075.15009

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.

MSC:

15A04 Linear transformations, semilinear transformations
15A03 Vector spaces, linear dependence, rank, lineability
15B33 Matrices over special rings (quaternions, finite fields, etc.)
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