A direct proof for the matrix decomposition of chordal-structured positive semidefinite matrices

Naonori Kakimura
2010 Linear Algebra and its Applications  
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.
doi:10.1016/j.laa.2010.04.012 fatcat:jkn36a2c4zaalh74wfho4l73pq