### Abstract

Given an affine subspace of square matrices, we consider the problem of minimizing the spectral abscissa (the largest real part of an eigenvalue). We give an example whose optimal solution has Jordan form consisting of a single Jordan block, and we show, using nonlipschitz variational analysis, that this behaviour persists under arbitrary small perturbations to the example. Thus although matrices with nontrivial Jordan structure are rare in the space of all matrices, they appear naturally in spectral abscissa minimization.

Original language | English (US) |
---|---|

Pages (from-to) | 1635-1642 |

Number of pages | 8 |

Journal | Proceedings of the American Mathematical Society |

Volume | 129 |

Issue number | 6 |

State | Published - 2001 |

### Fingerprint

### Keywords

- Eigenvalue optimization
- Jordan form
- Nonsmooth analysis
- Spectral abscissa

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Proceedings of the American Mathematical Society*,

*129*(6), 1635-1642.

**Optimizing matrix stability.** / Burke, J. V.; Lewis, A. S.; Overton, M. L.

Research output: Contribution to journal › Article

*Proceedings of the American Mathematical Society*, vol. 129, no. 6, pp. 1635-1642.

}

TY - JOUR

T1 - Optimizing matrix stability

AU - Burke, J. V.

AU - Lewis, A. S.

AU - Overton, M. L.

PY - 2001

Y1 - 2001

N2 - Given an affine subspace of square matrices, we consider the problem of minimizing the spectral abscissa (the largest real part of an eigenvalue). We give an example whose optimal solution has Jordan form consisting of a single Jordan block, and we show, using nonlipschitz variational analysis, that this behaviour persists under arbitrary small perturbations to the example. Thus although matrices with nontrivial Jordan structure are rare in the space of all matrices, they appear naturally in spectral abscissa minimization.

AB - Given an affine subspace of square matrices, we consider the problem of minimizing the spectral abscissa (the largest real part of an eigenvalue). We give an example whose optimal solution has Jordan form consisting of a single Jordan block, and we show, using nonlipschitz variational analysis, that this behaviour persists under arbitrary small perturbations to the example. Thus although matrices with nontrivial Jordan structure are rare in the space of all matrices, they appear naturally in spectral abscissa minimization.

KW - Eigenvalue optimization

KW - Jordan form

KW - Nonsmooth analysis

KW - Spectral abscissa

UR - http://www.scopus.com/inward/record.url?scp=33646843419&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33646843419&partnerID=8YFLogxK

M3 - Article

VL - 129

SP - 1635

EP - 1642

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 6

ER -