Linear/additive preservers of ranks 2 and 4 on alternate matrix spaces over fields
A square matrix \(A\) over a field \(F\) is ``alternate'' if it is skew-symmetric and all diagonal entries are \(0\) (the latter condition is implied by the former condition except when \text{char}\((F)=2\)). The \(F\)-space of all \(n\times n\) alternate matrices over \(F\) is denoted by \(K_{n}(F)\). When \(n=4\) we define \(q\) to be the linear transformation \(q\) of \(K_{4}(F)\) which interchanges the \((1,4)\)th and \((4,1)\)th entries of \(A\) with the \((2,3)\)th and \((3,2)\)th entries, respectively. The main results of this paper are the following. If \(f\) is a linear operator on \(K_{n}(F)\) with \(n\geq4\), then \(f\) preserves the ranks of matrices of ranks \(2\) or \(4\) if and only if there exists \(\gamma\in F\) and a nonsingular \(S\in M_{n}(F)\) such that either (i) \(f(A)=\gamma SAS^{T}\) for all \(A\); or (ii) \(n=4\) and \(f(A)=\gamma Sq(A)S^{T}\) for all \(A\). Moreover, provided \(F\) is not isomorphic to one of its own proper subfields, a similar conclusion holds under the weaker assumption that \(f\) is only an additive map. In this case \(f\) preserves the ranks of matrices of ranks \(2\) and \(4\) if and only if \(f\) is a composite of a field automorphism followed by a function of type (i) or (ii).
scientific article; zbMATH DE number 2119125