The completion problem for N-matrices

Abstract

An $n\times m$ matrix is called an $N$-matrix if all principal minors are negative. In this paper, we are interested in $N$-matrix completion problems, that is, when a partial $N$-matrix has an $N$-matrix completion. In general, a combinatorially or non-combinatorially symmetric partial $N$-matrix does not have an $N$-matrix completion. Here, we prove that a combinatorially symmetric partial $N$-matrix has an $N$-matrix completion if the graph of its specified entries is a 1-chordal graph. We also prove that there exists an $N$-matrix completion for a partial $N$-matrix whose associated graph is an undirected cycle.