Sparse spectrally arbitrary patterns

Abstract

Abstract. We explore combinatorial matrix patterns of order n for which some matrix entries are necessarily nonzero, some entries are zero, and some are arbitrary. In particular, we are interested in when the pattern allows any monic characteristic polynomial with real coefficients, that is, when the pattern is spectrally arbitrary. We describe some order n patterns that are spectrally arbitrary. We show that each superpattern of a sparse companion matrix pattern is spectrally arbitrary. We determine all the minimal spectrally arbitrary patterns of order 2 and 3. Finally, we demonstrate that there exist spectrally arbitrary patterns for which the nilpotent-Jacobian method fails.

Refined Inertia of Matrix Patterns

“…The next result describes the irreducible spectrally arbitrary patterns of order 3 as demonstrated in [10].…”

“…A useful technique is the nilpotent-centralizer method introduced in [11]. While this technique was introduced for sign patterns, it also applies to the zero patterns we have been discussing, as noted in [10] and described in the next theorem. A matrix N is nilpotent if N k = 0 for some positive integer k. A nilpotent matrix has index k if k is the smallest positive integer such that N k = 0.…”

“…The paper by Cavers and Fallat [3] reviews some of these results and also sets them in a more general setting, initiating the study of other types of patterns. Recently, there has started to be some consideration of zero-nonzero patterns, patterns with prescribed zero entries, nonzero entries and entries that are unrestricted (see for example, [10]). The concept of a refined inertially arbitrary pattern has also recently been defined in [8] in the context of nonzero patterns.…”