The N-matrix completion problem under digraphs assumptions

Abstract

An $n \times n$ matrix is called an $N$--matrix if all principal minors are negative. In this paper, we are interested in the partial $N$--matrix completion problem, when the partial $N$--matrix is non-combinatorially symmetric. In general, this type of partial matrices does not have an $N$--matrix completion. We prove that a non-combinatorially symmetric partial $N$--matrix has an $N$--matrix completion if the graph of its specified entries is an acyclic graph or a cycle. We also prove that there exists the desired completion for partial $N$--matrices such that in its associated graphs the cycles play an important role.