Additive preservers of rank-additivity on matrix spaces Original Research Article

Let image be a field. Let image denote the vector space of all m × n matrices over image or the vector space of all n × n symmetric matrices over image of characteristic not 2 or 3. For each fixed positive integer s greater-or-equal, slanted 2, let Qs denote the set of all matrix pairs (A, B) in image such that rank(A + B) = rank(A) + rank(B) less-than-or-equals, slant s. We characterize additive mappings ψ on image such that (ψ(A), ψ(B)) set membership, variant Qs whenever (A, B) set membership, variant Qs for a fixed s. We also describe the structure of linear mappings from the space of n × n matrices over image to the space of p × q matrices over image that preserve rank-additivity, where image.