Spectrally arbitrary pattern extensions

A matrix pattern is often either a sign pattern with entries in {0,+,−} or, more simply, a nonzero pattern with entries in {0,⁎}. A matrix pattern A is spectrally arbitrary if for any choice of a real matrix spectrum, there is a real matrix having the pattern A and the chosen spectrum. We describe a graphical technique, a triangle extension, for constructing spectrally arbitrary patterns out of some known lower order spectrally arbitrary patterns. These methods provide a new way of viewing some known spectrally arbitrary patterns, as well as providing many new families of spectrally arbitrary patterns. We also demonstrate how the technique can be applied to certain inertially arbitrary patterns to obtain larger inertially arbitrary patterns. We then provide an additional extension method for zero-nonzero patterns, patterns with entries in {0,⁎,⊛}.