Completions of P-matrix patterns (Q5926393)

A partially specified matrix is said to be a partial \(P\)-matrix if every fully specified principal submatrix is a \(P\)-matrix (i.e. a matrix such that all the determinants of its principal submatrices are positive. A pattern (a list of positions in the matrix) has a \(P\)-completion if every partial \(P\)-matrix specified by the pattern can be completed to a \(P\)-matrix.NEWLINENEWLINENEWLINEThe \(P\)-matrix completion problem was studied for the first time by \textit{C. R. Johnson} and \textit{B. K. Kroschel} [Electron. J. Linear Algebra 1, 59-63 (1996; Zbl 0889.15007)]. The present paper extends this work by providing a larger class of patterns having \(P\)-completion, including the \(4\times 4\) patterns with 8 or fewer off-diagonal positions. Also, an example from the first paper is extended to a family of patterns which do not have \(P\)-completion.NEWLINENEWLINENEWLINEThese results are applied to classify 207 of 218 patterns for \(4\times 4\) matrices as having \(P\)-completion or not having \(P\)-completion.