This article concerns weak monotonicity of matrices, with specific emphasis on its relationship with a certain class of proper splittings. The matrix A ∈ ℝ m×n is weak monotone provided Ax ≥ 0 ⇒ x ∈ ℝ n + +N(A), where N(A) is the nullspace of A. In particular, the following extension of well known characterizations forM-matrices is obtained. Suppose that int(ℝ m +){double intersection}ℝ(A) ≠φ. Then the statements (a) A is weak-monotone. (b) ℝ m + ∩ ℝ(A) ⊆ Aℝ n +. (c) There exists x 0 ≥ 0 such that Ax 0 > 0. satisfy (a) ⇔ (b) ⇒ (c). Suppose further that A can be written as A = U -V, where A and U have the same range space and null space, U and V are nonnegative, V U † ≥ 0 (where U † denotes the Moore-Penrose inverse of U), and Ax ≥ 0, Ux ≥ 0 =⇒ x ∈ ℝ n + + N(A). Then each of the above statements is equivalent to the statement (d) ρ(V U †) < 1.