#### Title

Matrices with Identical Sets of Neighbors

#### Document Type

Discussion Paper

#### Publication Date

5-1-1996

#### CFDP Number

1127

#### CFDP Pages

12

#### Abstract

Given a generic m by n matrix A , a lattice point h in Z is a neighbor of the origin if the body { x : Ax < b }, with b i = max{0, a i h }, i = 1, …, m , contains no lattice point other than 0 and h . The set of neighbors, N ( A ), is ﬁnite and Asymmetric. We show that if A’ is another matrix of the same size with the property that sign a i h = sign a i ’ h for every i and every h in N ( A ), then A’ has precisely the same set of neighbors as A . The collection of such matrices is a polyhedral cone, described by a ﬁnite set of linear inequalities, each such inequality corresponding to a generator of one of the cones C i = pos( h in N ( A ): a i h < 0}. Computational experience shows that C i has “few” generators. We demonstrate this in the ﬁrst nontrivial case n = 3, m = 4.

#### Recommended Citation

Bárány, Imre and Scarf, Herbert E., "Matrices with Identical Sets of Neighbors" (1996). *Cowles Foundation Discussion Papers*. 1373.

https://elischolar.library.yale.edu/cowles-discussion-paper-series/1373