Dissertation
Combinatorial properties of nonnegative and eventually nonnegative matrices
Washington State University
Doctor of Philosophy (PhD), Washington State University
08/2008
DOI:
https://doi.org/10.7273/000005805
Abstract
Nonnegative and eventually nonnegative matrices are useful in many areas of mathematics and have been widely studied. Perhaps the most famous results pertaining to nonnegative matrices are due to Perron and Frobenius and have prompted much further research. In this thesis, we investigate the combinatorial structure of nonnegative and eventually nonnegative matrices as it relates to the (peripheral) Jordan form of the matrices. Information obtained from the level form of a nonnegative matrix is used to gain insight into the Jordan form of the matrix, and vice-versa. In this paper, we give necessary and sufficient conditions for a set of Jordan blocks to correspond to the peripheral spectrum of a nonnegative matrix. For each eigenvalue, [lambda], the [lambda]-level characteristic (with respect to the spectral radius) is defined. The necessary and sufficient conditions include a requirement that the [lambda]-level characteristic is majorized by the [lambda]-height characteristic. An algorithm which has been implemented in MATLAB is given to determine when a multiset of Jordan blocks corresponds to the peripheral spectrum of a nonnegative matrix. We also consider the Jordan form of an eventually nonnegative matrix. It is known that the necessary and sufficient conditions for a multiset of Jordan blocks to correspond to the peripheral Jordan form of an eventually nonnegative matrix coincide exactly with the necessary and sufficient conditions for a multiset of Jordan blocks to correspond to the peripheral Jordan form of a nonnegative matrix. We take a closer look at the Jordan blocks associated with eigenvalues of smaller magnitude. Multiple sufficient conditions on the Jordan form of an (eventually) nonnegative matrix are given. We also offer necessary and sufficient conditions on the Jordan form of an eventually nonnegative matrix.
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Details
- Title
- Combinatorial properties of nonnegative and eventually nonnegative matrices
- Creators
- Deanne Morris
- Contributors
- Judith Joanne McDonald (Chair)
- Awarding Institution
- Washington State University
- Academic Unit
- Department of Mathematics and Statistics
- Theses and Dissertations
- Doctor of Philosophy (PhD), Washington State University
- Publisher
- Washington State University
- Number of pages
- 97
- Identifiers
- 99901055038301842
- Language
- English
- Resource Type
- Dissertation