### Abstract

A symmetric matrix A is said to be sign-nonsingular if every symmetric matrix with the same sign pattern as A is nonsingular. Hall, Li and Wang showed that the inertia of a sign-nonsingular symmetric matrix is determined uniquely by its sign pattern. The purpose of this paper is to present an efficient algorithm for computing the inertia of such symmetric matrices. The algorithm runs in O(√nm log n) time for a symmetric matrix of order n with m nonzero entries. In addition, it is shown to be NP-complete to decide whether the inertia of a given symmetric matrix is not determined by its sign pattern.

Original language | English |
---|---|

Pages (from-to) | 229-244 |

Number of pages | 16 |

Journal | Mathematical Programming |

Volume | 110 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2007 Jun |

Externally published | Yes |

### Fingerprint

### Keywords

- Inertia
- Sign patterns
- Sign-nonsingular symmetric matrices

### ASJC Scopus subject areas

- Software
- Mathematics(all)

### Cite this

*Mathematical Programming*,

*110*(1), 229-244. https://doi.org/10.1007/s10107-006-0056-6

**Computing the inertia from sign patterns.** / Kakimura, Naonori; Iwata, Satoru.

Research output: Contribution to journal › Article

*Mathematical Programming*, vol. 110, no. 1, pp. 229-244. https://doi.org/10.1007/s10107-006-0056-6

}

TY - JOUR

T1 - Computing the inertia from sign patterns

AU - Kakimura, Naonori

AU - Iwata, Satoru

PY - 2007/6

Y1 - 2007/6

N2 - A symmetric matrix A is said to be sign-nonsingular if every symmetric matrix with the same sign pattern as A is nonsingular. Hall, Li and Wang showed that the inertia of a sign-nonsingular symmetric matrix is determined uniquely by its sign pattern. The purpose of this paper is to present an efficient algorithm for computing the inertia of such symmetric matrices. The algorithm runs in O(√nm log n) time for a symmetric matrix of order n with m nonzero entries. In addition, it is shown to be NP-complete to decide whether the inertia of a given symmetric matrix is not determined by its sign pattern.

AB - A symmetric matrix A is said to be sign-nonsingular if every symmetric matrix with the same sign pattern as A is nonsingular. Hall, Li and Wang showed that the inertia of a sign-nonsingular symmetric matrix is determined uniquely by its sign pattern. The purpose of this paper is to present an efficient algorithm for computing the inertia of such symmetric matrices. The algorithm runs in O(√nm log n) time for a symmetric matrix of order n with m nonzero entries. In addition, it is shown to be NP-complete to decide whether the inertia of a given symmetric matrix is not determined by its sign pattern.

KW - Inertia

KW - Sign patterns

KW - Sign-nonsingular symmetric matrices

UR - http://www.scopus.com/inward/record.url?scp=33947167897&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33947167897&partnerID=8YFLogxK

U2 - 10.1007/s10107-006-0056-6

DO - 10.1007/s10107-006-0056-6

M3 - Article

AN - SCOPUS:33947167897

VL - 110

SP - 229

EP - 244

JO - Mathematical Programming

JF - Mathematical Programming

SN - 0025-5610

IS - 1

ER -