The P_0-matrix completion problem

Abstract

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.

The nonnegative P_0-matrix completion problem

“…In [6,Theorem 8.4], it is noted that the same method of proof applies to partial nonnegative P 0,1 -matrices. In contrast, it is shown in [2] that the analogous statement for P 0 -matrices is true if and only if n = 4. That is, a pattern for n×n matrices that includes all diagonal positions whose patterndigraph is a symmetric n-cycle has P 0 -completion if and only if n = 4.…”

“…In Section 3 it is shown that the same situation holds for the nonnegative P 0 -matrix completion problem, that is, a pattern for n × n matrices that includes all diagonal positions whose pattern digraph is a symmetric n-cycle has nonnegative P 0 -completion if and only if n = 4. Section 2 contains a classification of patterns for 4×4 matrices that include all diagonal positions as either having nonnegative P 0 -completion or not having nonnegative P 0 -completion, as [2] provided the analogous classification for P 0 -completion. However, this classification has a more elegant description (cf.…”

“…Theorem 2.6). In [2], it was established that asymmetric patterns have P 0 -completion. This is not the case for nonnegative P 0 -completion, as Lemma 2.1 makes clear.…”

The nonnegative Q−matrix completion problem

“…A Π-completion of a partial Π-matrix M is a completion of M which is a Π-matrix. Matrix completion problems for several classes of matrices including the classes of P and P 0 -matrices have been studied by a number of authors (e.g., [2,3,5,7,8,10,11]). …”

The matrix completion problem regarding various classes of P0,1- matrices

“…1 DetA and 3x3 principal minors of A in Case 1 Table 4.1 DetA and 3x3 principal minors of A in Case 1 Table 4.2 DetA and 3x3 principal minors of A in Case 2 with e = 0 23 Table 4. 3 DetA and 3x3 principal minors of A in Case 3A Table 4. 4 DetA and 3x3 principal minors of A in Case 3B Table 4.…”