Decompositions of minimum rank matrices

Abstract

Let F be a field, let G be an undirected graph on n vertices, and let SF(G) be the class of all F-valued symmetric n×n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. For each graph G, there is an associated minimum rank class MRF(G) consisting of all matrices A∈SF(G) with rankA=mrF(G). For most graphs G with connectivity 1 or 2, we give explicit decompositions of matrices in MRF(G) into sums of minimum rank matrices of simpler graphs (usually proper subgraphs) related to G. Our results can be thought of as generalizations of well-known formulae for the minimum rank of a graph with a cut vertex and of a graph with a 2-separation. We conclude by also showing that for these graphs, matrices in MRF(G) can be constructed from matrices of simpler graphs; moreover, we give analogues for positive semidefinite matrices.