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- W840804929 abstract "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) 6 �. Then the statements (a) A is weak-monotone. (b) R m ∩ R(A) ⊆ AR n . (c) There exists x0 ≥ 0 such that Ax0 > 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) �(V U † ) < 1." @default.
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- W840804929 date "2012-01-01" @default.
- W840804929 modified "2023-09-25" @default.
- W840804929 title "Weak monotonicity of matrices and subclasses of proper splittings" @default.
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- W840804929 doi "https://doi.org/10.13001/1081-3810.1581" @default.
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