Rank decompositions and signed bigraphs

Abstract

A signed bipartite graph G with vertices 1^′, 2^′,…,m and 1^′,2^′,…n^′, determines the family M(G) consisting of all m by n matrices whose (i,j)-entry is zero if i,j^′ )is not and edge of G nonnegative if {i,j^′} is an edge of G with label +1, and nonpositive if {i,f^′}is an edge of G with label -1. we show that each matrix A in M(G) can be expressed as the sum of rank(A) rank on matrices in M(G) if and only if

for every cycle γ of G of length l(γ)≥6. We also show that each matrix in M(G) has its rank equal to its term rank if and only if (1) holds for every cycle γ of G. Graphical characterizations of the signed bigraphs whose cycles satisfy (1) and of the signed bigraphs whose cycles of length 6 or more satisfy (1) are given.

^∗Supported in part by the Natural Sciences and Engineering Research Council of Canada under grant OGP0005134.

^†This research was done while this author was visiting the Department of Mathematics and Statistics at Queen's University, and is supported in part by the National Security Agency of U.S. under grant MDA904-94-H-2051.

^∗Supported in part by the Natural Sciences and Engineering Research Council of Canada under grant OGP0005134.

^†This research was done while this author was visiting the Department of Mathematics and Statistics at Queen's University, and is supported in part by the National Security Agency of U.S. under grant MDA904-94-H-2051.

Notes

^∗Supported in part by the Natural Sciences and Engineering Research Council of Canada under grant OGP0005134.

^†This research was done while this author was visiting the Department of Mathematics and Statistics at Queen's University, and is supported in part by the National Security Agency of U.S. under grant MDA904-94-H-2051.