Matrices attaining the minimum semidefinite rank of a chordal graph

Every positive semidefinite matrix whose graph G is chordal, may be represented as a sum of rank 1 positive semidefinite matrices whose graphs are subgraphs of G. We show that if the matrix is of minimum rank, then this representation is unique. This result is used to give a full characterization of the completely positive matrices of minimum rank with a given chordal graph. Every such matrix is shown to have a unique, easily computable, minimal rank 1 representation, and cp-rank equal to its rank. We also characterize all chordal graphs with the property that every minimum rank doubly nonnegative matrix realization of the graph is completely positive.