Spectrally arbitrary patterns over finite fields
| Autor: | E.J. Bodine, Judi J. McDonald |
|---|---|
| Rok vydání: | 2012 |
| Předmět: | |
| Zdroj: | Linear and Multilinear Algebra. 60:285-299 |
| ISSN: | 1563-5139 0308-1087 |
| Popis: | An n × n zero–nonzero pattern 𝒜 is spectrally arbitrary over a field 𝔽 provided that for each monic polynomial r(x)∈𝔽[x] of degree n, there exists a matrix A over 𝔽 with zero–nonzero pattern 𝒜 such that the characteristic polynomial p A (x) = r(x). In this article, we investigate several classes of zero–nonzero patterns over finite fields and algebraic extensions of ℚ. We prove that there are no spectrally arbitrary patterns over 𝔽2 and show that the full 2 × 2 pattern is spectrally arbitrary over 𝔽 if and only if F contains at least five elements. We explore an n × n pattern with precisely 2n nonzero entries that is spectrally arbitrary over finite fields 𝔽 q with , as well as ℚ. We also investigate an interesting 3 × 3 pattern for which the algebraic structure of the finite field rather than just the size of the field is a critical factor in determining whether or not it is spectrally arbitrary. This pattern turns out to be spectrally arbitrary over |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|