The \(P_0\)-matrix completion problem (Q2778502)

A partial matrix is a rectangular array in which some entries are specified while others are free to be chosen. A completion of a partial matrix is a specific choice of values for the unspecified entries. A pattern for \(n \times n\) matrices is a list of positions of the matrix.NEWLINENEWLINENEWLINEAn \(n \times n\) matrix is called a \(P_0\)-matrix (respectively, a \(P\)-matrix) if all the determinant of its principal submatrices are nonnegative (respectively, positive). A partial \(P_0\)-matrix (respectively, a partial \(P\)-matrix) is a partial matrix in which all fully specified principal submatrices are \(P_0\)-matrices (respectively, \(P\)-matrices).NEWLINENEWLINENEWLINEA partial matrix is called asymmetric if whenever \(i \neq j\) and \(a_{ij}\) is specified, then \(a_{ji}\) is not specified. A non-symmetric pattern for \(n \times n\) matrices that includes all diagonal positions can be described by means of a digraph on \(n\) vertices; namely, the directed edge \((i,j)\), \(1 \leq i,j \leq n\), is in the arc set of the digraph if and only if the ordered pair \((i,j)\) is in the pattern.NEWLINENEWLINENEWLINEThe authors prove that every asymmetric partial \(P\)-matrix (or \(P_0\)-matrix) has \(P\)-completion (or \(P_0\)-completion). Furthermore they prove that a pattern that includes all diagonal positions and whose digraph is a symmetric \(n\)-cycle has \(P_0\)-completion for \(n \geq 5\).NEWLINENEWLINENEWLINETables and examples support the results.