A new approach to some problems in the theory of non-negative matrices
Štefan Schwarz · Czechoslovak Mathematical Journal · 1966
In the paper [11] I developed a semigroup treatment of some theorems concerning non-negative matrices.The substance of this method is the follovv^ing.Denote iV = {1, 2, ..., n} and consider the set of all "w x n matrix units," i.e. the set of symbols {е^ \i eN,j eN} together with a zero 0 adjoined.Define in .S = = {0} u [eij \i eN, j eN} 3L multiplication by e..e ,=/^^^ ^""^ ^*^'^' the zero element having the usual properties of a multiplicative zero.The set S with this multiplication is a 0-simple semigroup containing n non-zero idempotents Let Ä = (uij) be a non-negative n x n matrix.By the support C^ of Л we shall mean the subset of S containing 0 and all e^ for which a^^ > 0.