The Nilpotent-Centralizer Methods

Abstract:

      An $n \times n$ complex sign pattern (ray pattern) ${\cal S}$ is said to be spectrally arbitrary if for every monic $n$th degree polynomial $f(\lambda)$ with coefficients from $\mathbb{C}$, there is a complex matrix in the complex sign pattern class (ray pattern class) of ${\cal S}$ such that its characteristic polynomial is $f(\lambda)$. We derive the Nilpotent-Centralizer methods for spectrally arbitrary complex sign patterns and ray patterns, respectively. We find that the Nilpotent-Centralizer methods for three kinds of patterns (sign pattern, complex sign pattern, ray pattern) are the same in form.