Linear operators that preserve pairs of matrices which satisfy extreme rank properties––a supplementary version

A pair of m×n matrices (A,B) is said to be rank-sum-maximal if ρ(A+B)=ρ(A)+ρ(B), and rank-sum-minimal if ρ(A+B)=ρ(A)−ρ(B). We characterize the linear operators preserving the set of rank-sum-maximal pairs over any field and the linear operators preserving the set of rank-sum-minimal pairs over any field except for {0,1}. The linear preservers of the set of rank-sum-maximal pairs are characterized by using a result about rank preservers proposed by Li and Pierce [Amer. Math. Monthly 108 (2001) 591–605], and thereby the linear preservers of the set of rank-sum-minimal pairs are characterized. The paper can be viewed as a supplementary version of several related results.