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

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8 Citations (Scopus)

Abstract

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 chordal-structured matrix due to Grone, Johnson, Sá, and Wolkowicz. 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.

Original languageEnglish
Pages (from-to)819-823
Number of pages5
JournalLinear Algebra and Its Applications
Volume433
Issue number4
DOIs
Publication statusPublished - 2010 Oct 1

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Keywords

  • Chordal graphs
  • Matrix completion problem
  • Positive semidefinite matrices

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Numerical Analysis
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics

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