On G-majorization of vectors

A real matrix [Formula: see text] is row generalized stochastic if the sum of absolute value of entries in each row is less than or equal to one and there is a fixed [Formula: see text] such that the sum of all entries in each row is [Formula: see text]. For vectors [Formula: see text] (respectively, [Formula: see text]), it is said that [Formula: see text] is right (respectively, left) generalized matrix majorized by [Formula: see text] (denoted by [Formula: see text] (respectively, [Formula: see text])) if [Formula: see text] (respectively, [Formula: see text]) for some [Formula: see text]-by-[Formula: see text] row generalized stochastic matrix [Formula: see text]. In this paper, we have found all vectors such as [Formula: see text] that are right-majorized by [Formula: see text] and we have found all vectors such as [Formula: see text] that are left-majorized by [Formula: see text] and we describe all equivalence classes [Formula: see text] and [Formula: see text].