Dissertation
Spectrally arbitrary patterns of matrices over finite fields
Washington State University
Doctor of Philosophy (PhD), Washington State University
08/2010
DOI:
https://doi.org/10.7273/000006113
Abstract
A zero-nonzero pattern A is spectrally arbitrary over a finite field Fq provided that for
each monic polynomial r(x) ∈ Fq[x], there exists a matrix A over Fq with zero-nonzero
pattern A such that the characteristic polynomial pA(x) = r(x). We investigate zero-nonzero patterns over finite fields and observe when particular patterns are spectrally arbitrary over a finite field. We prove that there are no spectrally arbitrary patterns over F2 and show that the full 2 × 2 pattern is spectrally arbitrary over Fq if and only if q ≥ 5.
The 3 × 3 patterns are fully characterized, leading to interesting observations and
conjectures. In particular, we examine the spectrally arbitrary zero-nonzero patterns
with the fewest number of nonzero entries possible, and make observations regarding
the results when there are additional nonzero entries. Particular types of 4×4 patterns are examined and grouped according to similarities in the structure of the pattern and the results. Some of these patterns are spectrally arbitrary over R, and simply require the identification of an appropriate lower bound on the field order. Other patterns explored herein are not spectrally arbitrary over R, but are spectrally arbitrary over C. We focus on the particular algebraic structure required of Fq in order for the patterns to be spectrally arbitrary over Fq. There are collections of patterns that require particular roots of unity, while others require that the characteristic not be 2. We explore several classes of n×n patterns and identify conditions for these patterns to be spectrally arbitrary. In some cases, this is just an issue of identifying a lower bound on the field order; for example we examine an n × n pattern that is spectrally arbitrary over finite fields Fq with q ≥ n(n+1) 2 + 1. In other cases, the requirements are more complicated. We specifically examine the tridiagonal pattern Tn and the n−cycle with loops Dn, and identify necessary criteria the field must meet in order for these patterns to be spectrally arbitrary. Many of these results over the finite fields can also be modified to lead to knowledge about spectrally arbitrary patterns over Q and finite extensions of Q. This is treated in the final section of the paper.
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Details
- Title
- Spectrally arbitrary patterns of matrices over finite fields
- Creators
- Elizabeth J. Bodine
- Contributors
- Judith Joanne McDonald (Chair)Michael Tsatsomeros (Committee Member) - Washington State University, Department of Mathematics and StatisticsMatthew G Hudelson (Committee Member) - Washington State University, Department of Mathematics and Statistics
- 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
- 126
- Identifiers
- 99901055119301842
- Language
- English
- Resource Type
- Dissertation