Additive operators preserving rank-additivity on symmetry matrix spaces

Abstract

We characterize the additive operators preserving rank-additivity on symmetry matrix spaces. Let Sn (F) be the space of all n × n symmetry matrices over a field F with 2, 3 ∈ F*, then T is an additive injective operator preserving rank-additivity on Sn (F) if and only if there exists an invertible matrix U ∈ Mn(F) and an injective field homomorphism φ of F to itself such that T(X) = cUXφUT, ∀X = (xij) ∈ Sn(F) where c ∈ F*, Xφ = (φ(xij)). As applications, we determine the additive operators preserving minus-order on Sn(F) over the field F.