### Abstract

Original language | English |
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Pages (from-to) | 403-414 |

Journal | Numerical Linear Algebra with Applications |

Volume | 21 |

Issue number | 3 |

Early online date | 30 May 2013 |

DOIs | |

Publication status | Published - May 2014 |

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### Cite this

*Numerical Linear Algebra with Applications*,

*21*(3), 403-414. https://doi.org/10.1002/nla.1888

**The calculation of the distance to a nearby defective matrix.** / Akinola, R. O.; Freitag, M. A.; Spence, A.

Research output: Contribution to journal › Article

*Numerical Linear Algebra with Applications*, vol. 21, no. 3, pp. 403-414. https://doi.org/10.1002/nla.1888

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TY - JOUR

T1 - The calculation of the distance to a nearby defective matrix

AU - Akinola, R. O.

AU - Freitag, M. A.

AU - Spence, A.

PY - 2014/5

Y1 - 2014/5

N2 - The distance of a matrix to a nearby defective matrix is an important classical problem in numerical linear algebra, as it determines how sensitive or ill-conditioned an eigenvalue decomposition of a matrix is. The concept has been discussed throughout the history of numerical linear algebra, and the problem of computing the nearest defective matrix first appeared in Wilkinsons famous book on the algebraic eigenvalue problem. In this paper, a new fast algorithm for the computation of the distance of a matrix to a nearby defective matrix is presented. The problem is formulated following Alam and Bora introduced in (2005) and reduces to finding when a parameter-dependent matrix is singular subject to a constraint. The solution is achieved by an extension of the implicit determinant method introduced by Spence and Poulton in (2005). Numerical results for several examples illustrate the performance of the algorithm.

AB - The distance of a matrix to a nearby defective matrix is an important classical problem in numerical linear algebra, as it determines how sensitive or ill-conditioned an eigenvalue decomposition of a matrix is. The concept has been discussed throughout the history of numerical linear algebra, and the problem of computing the nearest defective matrix first appeared in Wilkinsons famous book on the algebraic eigenvalue problem. In this paper, a new fast algorithm for the computation of the distance of a matrix to a nearby defective matrix is presented. The problem is formulated following Alam and Bora introduced in (2005) and reduces to finding when a parameter-dependent matrix is singular subject to a constraint. The solution is achieved by an extension of the implicit determinant method introduced by Spence and Poulton in (2005). Numerical results for several examples illustrate the performance of the algorithm.

UR - http://www.scopus.com/inward/record.url?scp=84878357407&partnerID=8YFLogxK

UR - http://dx.doi.org/10.1002/nla.1888

U2 - 10.1002/nla.1888

DO - 10.1002/nla.1888

M3 - Article

VL - 21

SP - 403

EP - 414

JO - Numerical Linear Algebra with Applications

JF - Numerical Linear Algebra with Applications

SN - 1070-5325

IS - 3

ER -