A note on Ostrowski's Theorem

Abstract

In this note, a further extension of Ostrowski's Theorem, concerning mainly complex square irreducible matrices, is presented. Specifically, classes of irreducible matrices are determined for which the classical statement: "If for a matrix A = [a(ij)] is an element of C-nxn, n >= 2, relations vertical bar a(ij)vertical bar > (Sigma(n)(j=1)(,j not equal 1) vertical bar a(ij)vertical bar)(alpha) (Sigma(n)(j=1)(,j not equal 1) vertical bar a(ij)vertical bar)(1-alpha) are satisfied for all i is an element of {1, 2, ... , n} and for some alpha is an element of [0, 1], then, A is non-singular", can hold even if all the inequalities in it turn out to be equalities. (C) 2013 Elsevier Inc. All rights reserved.