### 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 matrices. The algorithm runs in O(nm) time for a symmetric matrix of order n with m nonzero entries. The correctness of the algorithm provides an alternative proof of the result by Hall et al. In addition, for a symmetric matrix in general, it is shown to be NP-complete to decide whether the inertia of the matrix is not determined by the sign pattern.

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

Pages (from-to) | 236-248 |

Number of pages | 13 |

Journal | Lecture Notes in Computer Science |

Volume | 3509 |

Publication status | Published - 2005 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Lecture Notes in Computer Science*,

*3509*, 236-248.

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

Research output: Contribution to journal › Article

*Lecture Notes in Computer Science*, vol. 3509, pp. 236-248.

}

TY - JOUR

T1 - Computing the inertia from sign patterns

AU - Kakimura, Naonori

AU - Iwata, Satoru

PY - 2005

Y1 - 2005

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 matrices. The algorithm runs in O(nm) time for a symmetric matrix of order n with m nonzero entries. The correctness of the algorithm provides an alternative proof of the result by Hall et al. In addition, for a symmetric matrix in general, it is shown to be NP-complete to decide whether the inertia of the matrix is not determined by the 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 matrices. The algorithm runs in O(nm) time for a symmetric matrix of order n with m nonzero entries. The correctness of the algorithm provides an alternative proof of the result by Hall et al. In addition, for a symmetric matrix in general, it is shown to be NP-complete to decide whether the inertia of the matrix is not determined by the sign pattern.

UR - http://www.scopus.com/inward/record.url?scp=24944529898&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=24944529898&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:24944529898

VL - 3509

SP - 236

EP - 248

JO - Lecture Notes in Computer Science

JF - Lecture Notes in Computer Science

SN - 0302-9743

ER -