Potentially stable and 5-by-5 spectrally arbitrary tree sign pattern matrices with all edges negative

Characterization of potentially stable sign pattern matrices has been a long-standing open problem. In this paper, we give some sufficient conditions for tree sign pattern matrices with all edges negative to allow a properly signed nest. We also characterize potentially stable star and path sign pattern matrices with all edges negative. We give a conjecture on characterizing potentially stable tree sign pattern matrices with all edges negative in terms of allowing a properly signed nest which is verified to be true for sign pattern matrices up to order 6. Finally, we characterize all 5-by-5 spectrally arbitrary tree sign pattern matrices with all edges negative.

Algebraic conditions and the sparsity of spectrally arbitrary patterns

Given a square matrix A, replacing each of its nonzero entries with the symbol * gives its zero-nonzero pattern. Such a pattern is said to be spectrally arbitrary when it carries essentially no information about the eigenvalues of A. A longstanding open question concerns the smallest possible number of nonzero entries in an n × n spectrally arbitrary pattern. The Generalized 2n Conjecture states that, for a pattern that meets an appropriate irreducibility condition, this number is 2n. An example of Shitov shows that this irreducibility is essential; following his technique, we construct a smaller such example. We then develop an appropriate algebraic condition and apply it computationally to show that, for n ≤ 7, the conjecture does hold for ℝ, and that there are essentially only two possible counterexamples over ℂ. Examining these two patterns, we highlight the problem of determining whether or not either is in fact spectrally arbitrary over ℂ. A general method for making this determination for a pattern remains a major goal; we introduce an algebraic tool that may be helpful.

Spectral Properties of Sign Patterns

In this paper, an infinite family of irreducible sign patterns that are spectrally arbitrary, for which the nilpotent-Jacobian method does not apply, is given. It is demonstrated that it is possible for an irreducible sign pattern to be refined inertially arbitrary and not spectrally arbitrary. It is observed that not every nonzero spectrally arbitrary pattern has a signing which is spectrally arbitrary. It is also shown that every superpattern of the reducible pattern $\T_2 \oplus \T_2$ is spectrally arbitrary.

Refined Inertia of Matrix Patterns

This paper explores how the combinatorial arrangement of prescribed zeros in a matrix affects the possible eigenvalues that the matrix can obtain. It demonstrates that there are inertially arbitrary patterns having a digraph with no 2-cycle, unlike what happens for nonzero patterns. A class of patterns is developed that are refined inertially arbitrary but not spectrally arbitrary, making use of the property of a properly signed nest. The paper includes a characterization of the inertially arbitrary and refined inertially arbitrary patterns of order three, as well as the patterns of order four with the least number of nonzero entries.