Weak monotonicity of matrices and subclasses of proper splittings
Agarwal Sushama, K. Premakumari, K. C. Sivakumar · Electronic Journal of Linear Algebra · 2012
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 for M -matrices is obtained.Suppose that int(R m + )∩R(A) = φ.Then the statementssatisfy (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, U x ≥ 0 =⇒ x ∈ R n + + N (A).Then each of the above statements is equivalent to the statement