Let [Formula: see text] be a 2-torsion free unital generalized matrix algebra with center [Formula: see text], and [Formula: see text] be a linear mapping on [Formula: see text] satisfying the condition [Formula: see text] This paper is devoted to the study of the structure of [Formula: see text] under some mild assumptions on [Formula: see text]. We provide the necessary and sufficient conditions for [Formula: see text] to be in the form [Formula: see text] ([Formula: see text]), where [Formula: see text] and [Formula: see text] is a linear mapping. Then we apply our results to characterize linear mappings on [Formula: see text] that are commutant preservers or double commutant preservers.