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

Research output: Contribution to journalArticle

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
Externally publishedYes

Fingerprint

Matrix Decomposition
Positive Semidefinite Matrix
Positive semidefinite
Decomposition
Matrix Completion
Maximal Clique
Structured Matrices
Semidefinite Programming
Graph in graph theory
Symmetric matrix
If and only if
Decompose
Alternatives

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

Cite this

A direct proof for the matrix decomposition of chordal-structured positive semidefinite matrices. / Kakimura, Naonori.

In: Linear Algebra and Its Applications, Vol. 433, No. 4, 01.10.2010, p. 819-823.

Research output: Contribution to journalArticle

@article{ec7e6ced5c9141f99ac8f71a75af2b10,
title = "A direct proof for the matrix decomposition of chordal-structured positive semidefinite matrices",
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, S{\'a}, 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.",
keywords = "Chordal graphs, Matrix completion problem, Positive semidefinite matrices",
author = "Naonori Kakimura",
year = "2010",
month = "10",
day = "1",
doi = "10.1016/j.laa.2010.04.012",
language = "English",
volume = "433",
pages = "819--823",
journal = "Linear Algebra and Its Applications",
issn = "0024-3795",
publisher = "Elsevier Inc.",
number = "4",

}

TY - JOUR

T1 - A direct proof for the matrix decomposition of chordal-structured positive semidefinite matrices

AU - Kakimura, Naonori

PY - 2010/10/1

Y1 - 2010/10/1

N2 - 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.

AB - 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.

KW - Chordal graphs

KW - Matrix completion problem

KW - Positive semidefinite matrices

UR - http://www.scopus.com/inward/record.url?scp=77953133068&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77953133068&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2010.04.012

DO - 10.1016/j.laa.2010.04.012

M3 - Article

AN - SCOPUS:77953133068

VL - 433

SP - 819

EP - 823

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - 4

ER -