Positivity 2: 165–170, 1998.
© 1998 Kluwer Academic Publishers. Printed in the Netherlands.
On Positive Invertibility of matrices
Department of Mathematics, Ben Gurion University of the Negev, P.0. Box 653, Beer-Sheva 84105,
(Received: 7 April 1997; accepted in revised form: 20 February 1998)
Abstract. A new criterion for the positive invertibility of matrices is established. It improves the
theorem by Collatz for matrices which are “near” to triangular ones.
Mathematics Subject Classiﬁcation (1991): 15A18
Key words: matrices, positive invertibility
1. Introduction and Statement of the Result
Let A = (a
be a real n × n-matrix satisfying the conditions
≤ 0forj= k, and a
> 0 (j, k = 1,... ,n). (1.1)
> 0 for at least one j
Collatz has established the following result: under (1.1), let one of the follow-
ing conditions be satisﬁed: (a) A is nondecomposable and (1.2) holds, or (b) the
> 0forallj =1,... ,n (1.3)
hold. Then matrix A is monotone (positively invertible) [1, p. 377] (see also [4,
Section 25.4] ). That is, A is invertible and A
has non-negative entries. As ex-
amples show, conditions (1.2) and (1.3) are not necessary. For instance, let n = 2.
Then under (1.1) for the positive invertibility it is necessary and sufﬁcient that
In the present paper, new positive invertibility conditions are obtained. They are
more precise than conditions (1.2) and (1.3) for matrices which are “near” to
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