Craig Erickson and In-Jae Kim Minnesota State University and Minnesota State University
Abstract : The work in this paper was motivated by [3], where Eschenbach and Li listed four $4$ by $4$ sign patterns, conjectured to be nilpotent sign patterns of nilpotence index at least $3$. These sign patterns with no zero entries, called full sign patterns, are shown to be potentially nilpotent of nilpotence index $3$. We also generalize these sign patterns of order $4$ so that we provide classes of $n$ by $n$ sign patterns of nilpotence indices at least $3$, if they are potentially nilpotent. Furthermore it is shown that if a full sign pattern $\mathcal{A}$ of order $n$ has nilpotence index $k$ with $2 \leq k \leq n-1$, then sign pattern $\mathcal{A}$ has nilpotent realizations of nilpotence indices $k, k+1, \ldots, n$. Hence, the four $4$ by $4$ sign patterns in [3, page 91] also allow nilpotent realizations of nilpotence index $4$.
Keywords : Jordan block, nilpotence index, potentially nilpotent sign pattern
