Rank decompositions and signed bigraphs

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.