Minimal Spectrally Arbitrary Sign Patterns

An n × n sign pattern A is spectrally arbitrary if given any self-conjugate spectrum there exists a matrix realization of A with that spectrum. If replacing any nonzero entry (or entries) of A by zero destroys this property, then A is a minimal spectrally arbitrary sign pattern. For n ≥ 3, several families of n × n spectrally arbitrary sign patterns are presented, and their minimal spectrally arbitrary subpatterns are identified. These are the first known families of n × n minimal spectrally arbitrary sign patterns. Furthermore, all such 3 × 3 sign patterns are determined and it is proved that any irreducible n × n spectrally arbitrary sign pattern must have at least 2n − 1 nonzero entries, and conjectured that the minimum number of nonzero entries is 2n. 1. Introduction. A sign pattern is a square matrix with entries in {+, −, 0}. If A is a sign pattern and A is a real matrix for which each entry has the same sign as the corresponding entry of A, then A is said to be a realization of A, and we write A ∈ A. This convention is also used for zero-nonzero patterns A. A sign pattern B = [b ij ] is a superpattern of a sign pattern A = [a ij ] if b ij = a ij whenever a ij = 0. Similarly, B is a subpattern of A if b ij = 0 whenever a ij = 0. Note that each sign pattern is a superpattern and a subpattern of itself. An n × n sign pattern A is spectrally arbitrary if for each real monic polynomial r(x) of degree n, there exists some A ∈ A with characteristic polynomial p A (x) = r(x). Thus, A is spectrally arbitrary if given any self-conjugate spectrum, there exists A ∈ A with that spectrum. A sign pattern A is minimally spectrally arbitrary if it is spectrally arbitrary but is not spectrally arbitrary if any nonzero entry (or entries) of A is replaced by zero. If A is an n × n sign pattern or zero-nonzero pattern, then A allows nilpotency if there exists some A ∈ A with characteristic polynomial p A (x) = x n. Note that each spectrally arbitrary sign pattern must allow nilpotency, must be inertially arbitrary (as explained below Theorem 2.5), and must also be potentially stable. These are three …