Dissertation
Spectrally Arbitrary Zero-nonzero Patterns of Matrices Over a Variety of Fields
Doctor of Philosophy (PhD), Washington State University
01/2013
Handle:
https://hdl.handle.net/2376/5042
Abstract
A n × n zero-nonzero pattern A is spectrally arbitrary over a field F if for each monic, degree n polynomial p(t) in F[t], there exists a matrix A over F with zero-nonzero pattern A such that the characteristic polynomial of F is p(t).
The nilpotent-Jacobian method is the most widely used method to determine if a zero-nonzero pattern is spectrally arbitrary over the real and complex fields, but this method only works over topologically complete fields. We analyze the proof of the nilpotent-Jacobian to develop a method to determine if a zero-nonzero pattern is spectrally arbitrary over field extensions of the rational field, including Q itself. We also show that in certain situations the nilpotent-Jacobian method can be used over field extensions of Q. We then use these methods to classify all 2× 2 and 3× 3 patterns over extensions of Q.
We then characterize each real and complex spectrally arbitrary 4× 4 pattern over field extensions of Q. In doing so we observe that each real spectrally arbitrary pattern is spectrally arbitrary over the real algebraic closure of Q and each complex spectrally arbitrary pattern is spectrally arbitrary over the algebraic closure of Q. We prove that any complex spectrally arbitrary pattern will be spectrally arbitrary over the algebraic closure of Q, that is transcendental numbers are not needed to satisfy polynomials with algebraic coefficients. We conjecture that any real spectrally arbitrary pattern will be spectrally arbitrary over the real algebraic closure of Q.
A pattern B is a superpattern of a pattern A if aij=bij whenever aij is nonzero. The superpattern conjecture says that if a pattern is spectrally arbitrary over a field F then each of its superpatterns is spectrally arbitrary F. The superpattern conjecture is an open question over field extensions of Q, but we explore a number of counterexamples to the superpattern conjecture over the finite field of order 3.
The 2n conjecture states that 2n is the minimum number of nonzero entries of a spectrally arbitrary pattern. It has been proven that the minimum number of nonzero entries of a real or complex spectrally arbitrary pattern is 2n-1, and we prove 2n-1 is a lower bound of nonzero entries of a spectrally arbitrary pattern over any field extension of Q.
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Details
- Title
- Spectrally Arbitrary Zero-nonzero Patterns of Matrices Over a Variety of Fields
- Creators
- Timothy Christopher Melvin
- Contributors
- Judith J McDonald (Advisor)Michael Tsatsomeros (Committee Member)Matthew Hudelson (Committee Member)
- Awarding Institution
- Washington State University
- Academic Unit
- Mathematics and Statistics, Department of
- Theses and Dissertations
- Doctor of Philosophy (PhD), Washington State University
- Number of pages
- 120
- Identifiers
- 99900581739901842
- Language
- English
- Resource Type
- Dissertation