Eight types of matrix monotonicity

A real matrix A is called S-monotone if S is a complementary subspace of N(A) and if Ax⩾0, x∈S → x⩾0 . A is called almost monotone if it is S-monotone for every such S. These types of monotonicity appear in the study of regular splittings for iterative methods. Almost and S-monotonicity are characterized here and are related to six other types of monotonicity.