Minimal Spectrally Arbitrary Sign Patterns

Abstract

An $n\times n$ sign pattern $\mathcal{A}$ is spectrally arbitrary if given any self-conjugate spectrum there exists a matrix realization of $\mathcal{A}$ with that spectrum. If replacing any nonzero entry of $\mathcal{A}$ by zero destroys this property, then $\mathcal{A}$ is a minimal spectrally arbitrary sign pattern. Several families of sign patterns are presented that, for all $n\geq 3$, each contain an $n\times n$ minimal spectrally arbitrary sign pattern. These are the first families proven to have this property, and they improve previously known results. Furthermore, all $3\times 3$ minimal spectrally arbitrary sign patterns are determined, it is proved that any irreducible $n\times n$ spectrally arbitrary sign pattern must have at least $2n-1$ nonzero entries, and it is conjectured that the minimum number of nonzero entries is $2n$.