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For X, Y ∈ Mn,m it is said that X is gut-majorized by Y, and we write X ≺gutY, if there exists an n-by-n upper triangular g-row stochastic matrix R such that X = RY. Define the relation ~gut as follows. X ~gutY if X is gut-majorized by Y and Y is gut-majorized by X. The (strong) linear preservers of ≺gut on ℝn and strong linear preservers of this relation on Mn,m have been characterized before. This paper characterizes all (strong) linear preservers and strong linear preservers of ~gut on ℝn and Mn,m.

On linear preservers of two-sided gut-majorization on $ M_{n,m}$