Dissertation
Extensions of Nonnegative Matrices
Doctor of Philosophy (PhD), Washington State University
01/2019
Handle:
https://hdl.handle.net/2376/118072
Abstract
The Frobenius normal form of a matrix is an important tool in analyzing its properties. When a matrix is powered up, the Frobenius normal form of the original matrix and that of its powers need not be the same. In this dissertation, conditions on a matrix $A$ and the power $q$ are provided so that for any invertible matrix $S$, if $S^{-1}A^qS$ is block upper triangular, then so is $S^{-1}AS$ when partitioned conformably. The result is established for general matrices over any field. It is also observed that the contributions of the index of cyclicity to the spectral properties of a matrix hold over any field. We apply the block upper triangular powers result to the cone Frobenius normal form of powers of an eventually cone nonnegative matrix. Furthermore, we give a counterexample to the statement of Barker \\cite{barker1} Theorem 7, and extend the Barker theorem to a larger class of cones called orthogonal face free cone using the necessary and sufficient conditions on the closedness of the linear image of a closed convex cone. In addition, we provide the generators for a proper polyhedral cone ${\\cal K}$, such that $A$ is eventually ${\\cal K}$-irreducible with a small number of extreme vectors. We conclude this dissertation by a brief discussion of the general construction of Frobenius normal form with respect to cones which incorporate the idea of level sets.
Metrics
9 File views/ downloads
68 Record Views
Details
- Title
- Extensions of Nonnegative Matrices
- Creators
- Mashael Mothabet AlBaidani
- Contributors
- Judith McDonald (Advisor)Michael Tsatsomeros (Committee Member)Matthew Hudelson (Committee Member)
- Awarding Institution
- Washington State University
- Academic Unit
- Department of Mathematics and Statistics
- Theses and Dissertations
- Doctor of Philosophy (PhD), Washington State University
- Number of pages
- 82
- Identifiers
- 99900581708001842
- Language
- English
- Resource Type
- Dissertation