Spectrally arbitrary patterns over finite fields

Abstract

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

Keywords:

• zero–nonzero pattern
• spectrally arbitrary
• characteristic polynomial
• finite field

AMS Subject Classifications::

• 15A18, 15A33