Abstract. In this paper the P 0 -matrix completion problem is considered. It is established that every asymmetric partial P 0 -matrix has P 0 -completion. All 4 × 4 patterns that include all diagonal positions are classified as either having P 0 -completion or not having P 0 -completion. It is shown that any positionally symmetric pattern whose graph is an n-cycle with n ≥ 5 has P 0 -completion.
Abstract. In this paper it is shown that a partial sign symmetric P -matrix, whose digraph of specified entries is a symmetric n-cycle with n ≥ 6, can be completed to a sign symmetric Pmatrix. The analogous completion property is also established for a partial weakly sign symmetric P -matrix and for a partial weakly sign symmetric P 0 -matrix. Patterns of entries for 4 × 4 matrices are classified as to whether or not a partial (weakly) sign symmetric P -or weakly sign symmetric P 0 -matrix specifying the pattern must have completion to the same type of matrix. The relationship between the weakly sign symmetric P -and sign symmetric P -matrix completion problems is also examined.
A P-matrix is a real square matrix having every principal minor positive, and a Fischer matrix is a P-matrix that satisfies Fischer's inequality for all principal submatrices. In this paper, all patterns of positions for n × n matrices, n ≤ 4, are classified as to whether or not every partial Π-matrix can be completed to a Π-matrix for Π any of the classes positive P-, nonnegative P-, or Fischer matrices. Also, all symmetric patterns for 5 × 5 matrices are classified as to completion of partial Fischer matrices, and all but 2 such patterns are classified as positive Por nonnegative P-completion. We also show that any pattern whose digraph contains a minimally chordal symmetric-Hamiltonian induced subdigraph does not have Π-completion for Π any of the classes positive P-, nonnegative P-, Fischer matrices.
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