Rank decomposability in incident spaces (Q1904043)

The authors make the following definition: A set \({\mathcal M}\) of matrices is rank decomposable if each matrix \(T\) in \({\mathcal M}\) is the sum of \(r\) rank one matrices in \({\mathcal M}\), where \(r\) is the rank of \(T\). They show that an incidence space, i.e. the set of matrices supported on a given pattern, is rank decomposable if and only if the bipartite graph associated with the pattern is chordal.