Spectrally and inertially arbitrary sign patterns

We introduce some n-by-n sign patterns which allow for arbitrary spectrum and hence also arbitrary inertia. Consequently, we demonstrate that some known inertially arbitrary patterns are in fact spectrally arbitrary. We demonstrate that all inertially arbitrary patterns of order 3 are spectrally arbitrary and classify all spectrally arbitrary patterns of order 3. We illustrate that in general, the class of spectrally arbitrary patterns is distinct from the inertially arbitrary patterns, and present some observations about inertially arbitrary patterns.