The \(N\)-matrix completion problem under digraphs assumptions.
A real partial matrix is an \(n \times n\) array in which some entries are specified, while the remaining entries are free to be chosen. An \(n \times n\) partial matrix is said to be \textit{combinatorially symmetric} if the \((i,j)\) entry is specified if and only if the \((j,i)\) entry is specified, and non-combinatorially symmetric in the opposite case. A completion of a partial matrix is the matrix obtained after choosing the values for the unspecified entries. A matrix completion problem asks which partial matrices have completions with a given property. An \(n \times n \) real matrix is called N-matrix if all its principal minors are negative; these matrices occur in several mathematical problems. An \(n \times n \) real partial matrix is called partial N-matrix if every completely specified principal submatrix is an N-matrix. The authors study the N-matrix completion problem and the case of partial matrices \(A = (a_{ij})\) such that \(a_{ij}\neq 0\) and \(sign(a_{ij})=(-1)^{i+j+1}\) provided that the entry \((i,j)\) is specified. The specified entries of an \(n \times n \) partial matrix can be described by the digraph \((V,E)\): the set of vertices is \(V=\{1, \dots , n\}\) and the pair \((i,j) \;, \;i \neq j \), is an edge if and only if the \((i,j)\) entry is specified. The authors give an affirmative answer to the \(N\)-matrix completion problem in the case that the associated digraph contains no cycle and in the case that it contains particular kinds of cycles.
scientific article; zbMATH DE number 2062868