### 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 language | English |
---|---|

Pages (from-to) | 819-823 |

Number of pages | 5 |

Journal | Linear Algebra and Its Applications |

Volume | 433 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2010 Oct 1 |

Externally published | Yes |

### Fingerprint

### 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.

Research output: Contribution to journal › Article

}

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 -