Spectrally arbitrary complex sign pattern matrices

An n × n complex sign pattern matrix S is said to be spectrally arbitrary if for every monic nth degree polynomial f(λ) with coefficients from C, there is a complex matrix in the complex sign pattern class of S such that its characteristic polynomial is f(λ). If S is a spectrally arbitrary complex sign pattern matrix, and no proper subpattern of S is spectrally arbitrary, then S is a minimal spectrally arbitrary complex sign pattern matrix. This paper extends the NilpotentJacobian method for sign pattern matrices to complex sign pattern matrices, establishing a means to show that an irreducible complex sign pattern matrix and all its superpatterns are spectrally arbitrary. This method is then applied to prove that for every n ≥ 2 there exists an n×n irreducible, spectrally arbitrary complex sign pattern with exactly 3n nonzero entries. In addition, it is shown that every n × n irreducible, spectrally arbitrary complex sign pattern matrix has at least 3n − 1 nonzero entries.