The nonnegative P_0-matrix completion problem

Abstract

Abstract. In this paper the nonnegative P 0 -matrix completion problem is considered. It is shown that a pattern for 4 × 4 matrices that includes all diagonal positions has nonnegative P 0 -completion if and only if its digraph is complete when it has a 4-cycle. It is also shown that any positionally symmetric pattern that includes all diagonal positions and whose graph is an n-cycle has nonnegative P 0 -completion if and only if n = 4.

The nonnegative Q−matrix completion problem

“…P 1 -the loop at 3, φ * (P 1 ) = 1 P 2 -the union of the loops at 3 and 4, φ * (P 2 ) = 2 P 3 -the 3-cycle [1,4,2], φ * (P 3 ) = 3 P 4 -the union of the loop at 3 and the 3-cycle [1, 4, 2], φ * (P 4 ) = 4. Then D has nonnegative Q-completion.…”

“…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 (weakly) sign symmetric P-matrix completion problems

“…Complete B to C by Theorem 3.2 of [2]. Then DCD will complete A to a weakly sign symmetric P 0 -matrix.…”

“…Then A is a nonnegative partial P 0 -matrix and can be completed to a nonnegative (hence weakly sign symmetric) matrix [2].…”

“…We use most of the definitions and notation from [2]. One slight distinction: here the term "symmetric" is used for a pattern with the property that (j, i) is in the pattern whenever (i, j) is in the pattern; such patterns were called "positionally symmetric" in [2].…”