| We look at some of the results about the spectrum and the cyclicity of irreducible nonnegative matrices. These results are used in order to obtain information about the class of eventually nonnegative matrices. A matrix A is eventually nonnegative, if for all k sufficiently large, Ak is entry-wise nonnegative. Many of the properties associated with nonnegative matrices are not apparent in the eventually nonnegative matrices. Unlike nonnegative matrices, the cyclicity of an eventually nonnegative matrix is not tied to the cyclicity of its spectrum. One of the goals of this thesis is to develop an understanding of why and how this occurs.; We investigate cyclicity properties of Frobenius multisets and Frobenius collections of Jordan blocks.; We prove that all eventually nonnegative matrices are similar to seminonnegative matrices.; Using the Jordan bases for generalized eigenspaces of irreducible nonnegative blocks, we examine the reducibility of a decomposition of eventually nonnegative matrices. Let B be an eventually nonnegative matrix whose Jordan blocks corresponding to zero, if there are any, are all one by one. Using the Frobenius normal form for a certain prime power g and the union of Jordan bases for the generalized eigenspaces of the irreducible blocks of B, we demonstrate that B has the same Frobenius normal form as Bg. (Abstract shortened by UMI.)... |
