Agler, Helton, McCullough, and Rodman proved that a graph is chordal if and only if any positive semidefinite (PSD) symmetric matrix, whose nonzero entries are specified by a given graph, can be decomposed as a sum of PSD matrices corresponding to the maximal cliques. This decomposition is recently exploited to solve positive semidefinite programming efficiently. Their proof is based on a characterization for PSD matrix completion of a chordalstructured matrix due to Grone, Johnson, Sá, and

more »

... owicz. This note gives a direct and simpler proof for the result of Agler et al., which leads to an alternative proof of Grone et al.