The combinatorial structure of eventually nonnegative matrices

Abstract

In this paper it is shown that an eventuallynonnegative matrix A whose index of zero is less than or equal to one, exhibits manyof the same combinatorial properties as a nonnegative matrix. In particular, there is a positive integer g such that A g is nonnegative, A and A g have the same irreducible classes, and the transitive closure of the reduced graph of A is the same as the transitive closure of the reduced graph of Ag. In this instance, manyof the combinatorial properties of nonnegative matrices carryover to this subclass of the eventuallynonnegative matrices. 1. Introduction. The Perron-Frobenius Theorem for irreducible nonnegative matrices has spawned a wealth of interesting ideas in the study of nonnegative ma- trices. Graph-theoretic spectral theory of matrices continues to develop, and in this paper we are interested in extending many of these ideas to the class of eventually nonnegative matrices. The relationship between the combinatorial structure of a non- negative matrix and its spectrum, eigenvectors, and Jordan structure is surprisingly elegant and beautiful, as well as useful. Surveys of results of this type can be found in Berman and Plemmons (1), Hershkowitz (5), and Schneider (13). Friedland (3), Handelman (4), Zaslavsky and Tam (16), and Zaslavsky and Mc- Donald (15) have looked at extending some of these combinatorial ideas to eventually positive matrices and eventually nonnegative matrices. In their study they found examples of eventually nonnegative matrices for which the relationship between the combinatorial structure of the matrix, and its spectrum, eigenvectors, and Jordan structure was inconsistent with that of nonnegative matrices. For example, there are irreducible eventually nonnegative matrices for which the spectral radius is a multiple eigenvalue. Even when the spectral radius is a simple eigenvalue of an irreducible even- tually nonnegative matrix, the associated eigenvector need not be positive. Moving to properties of reducible eventually nonnegative matrices, we see that the combinatorial spectral properties exhibited by reducible nonnegative matrices need not carry over to eventually nonnegative matrices. In this paper we show that it is really the contributions from the nilpotent part of an eventually nonnegative matrix that determine whether or not the combinatorial structure will be an accurate predictor of its spectral properties. In Section 3, we show that if A is an eventually nonnegative matrix that is nonsingular, or if all the Jordan blocks in the Jordan form of A associated with the eigenvalue zero are 1 × 1, then there is a positive integer g such that the transitive closure of the reduced graph