More on generalizations of matrix monotonicity

AbstractFollowing Berman and Plemmons [5], Werner [17], Poole and Barker [13], and others, we investigate some generalizations of matrix monotonicity and consider their relation to MP matrices. We prove some observations on almost monotone, MP, and group monotone matrices. However, of much interest to us is the problem whether a symmetric positive semidefinite matrix is MP if and only if it is monotone on its range. To consider this question we obtain several results on the structure of range monotone matrices