Refined inertias of nonnegative patterns with positive off-diagonal entries

Abstract

For a positive $n\times n$ pattern $A$, it is known that the refined inertia of $A$, $\mathrm{ri}\left(A\right)$, is the set of all nonnegative integral 4-tuples $(n_{+},n_{-},n_z,2n_p)$ with $n_{+}+n_{-}+n_z+2n_p=n$ and $n_{+}\ge 1$; whereas if $A$ has all off-diagonal entries positive but all diagonal entries 0, then $\mathrm{ri}\left(A\right)$ has the additional restriction that $n_{-}\ge 2$. We focus on the intermediate nonnegative patterns, that is those patterns with all off-diagonal entries positive, $k\in \{1,\dots ,n-1\}$ diagonal entries positive and the remaining $n-k$ diagonal entries 0. We show that for $k\ge 2$, there is no restriction on $n_{-}$ for the refined inertia set, but $n_{-}\ge 1$ for $k=1$. We do this by constructing nonnegative matrix realizations for the patterns with $k=1$ and 2 using the centralizer method, matrix bordering and superpattern results.