Additive mappings on symmetric matrices
Let \(F_1\) and \(F_2\) be two fields and let \(S_n(F_1)\) and \(S_n(F_2)\) be the set of symmetric matrices over \(F_1\) and \(F_2\), respectively. A mapping \(\Phi: S_n(F_1)\to S_n(F_2)\) is called additive if \(\Phi(A+B)=\Phi(A)+\Phi(B)\). It is said that \(\Phi\) doesn't increase rank-one if \(rk(\Phi(A))\leq1\) whenever \(rk(A)=1\). \textit{M. H. Lim} [Linear Algebra Appl. 402, 263--271 (2005; Zbl 1084.15008)] has characterized the class of rank-\(1\) nonincreasing additive mappings on the symmetric matrices for the fields of characteristic \(\neq{2}\) and \(3\). The authors in the paper under review characterize all such additive mappings in characteristic \(2\) and \(3\). They use bases for matrix spaces and prove their results case by case. Although influenced by Lim [loc. cit.], they fill in the missing gaps in the mentioned paper.
scientific article; zbMATH DE number 5072868