| Popis: |
An n × n complex sign pattern matrix S is said to be spectrallyarbitraryif for everymonic 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 arbitrarycomplex sign pattern matrix, and no proper subpattern of S is spectrallyarbitrary , then S is a minimal spectrallyarbitrarycomplex sign pattern matrix. This paper extends the Nilpotent- Jacobian 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 ≥ 2t here exists ann × n irreducible, spectrallyarbitrarycomplex sign pattern with exactly3 n nonzero entries. In addition, it is shown that every n × n irreducible, spectrallyarbitrarycomplex sign pattern matrix has at least 3 n − 1 nonzero entries. |
