### Abstract

The ε-pseudospectral abscissa αε and radius ρε of an n×n matrix are, respectively, the maximal real part and the maximal modulus of points in its ε-pseudospectrum, defined using the spectral norm. It was proved in [A.S. Lewis and C.H.J. Pang, SIAM J. Optim., 19 (2008), pp. 1048-1072] that for fixed ε > 0, αε and ρε are Lipschitz continuous at a matrix A except when αε and ρε are attained at a critical point of the norm of the resolvent (in the nonsmooth sense), and it was conjectured that the points where αε and ρε are attained are not resolvent-critical. We prove this conjecture, which leads to the new result that αε and ρε are Lipschitz continuous, and also establishes the Aubin property with respect to both ε and A of the ε-pseudospectrum for the points z ε ℂ where αε and ρε are attained. Finally, we give a proof showing that the pseudospectrum can never be "pointed outwards."

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

Pages (from-to) | 281-285 |

Number of pages | 5 |

Journal | SIAM Journal on Optimization |

Volume | 22 |

Issue number | 2 |

DOIs | |

State | Published - 2012 |

### Fingerprint

### Keywords

- Aubin property
- Eigenvalue perturbation
- Lipschitz multifunction
- Pseudospectral abscissa
- Pseudospectral radius
- Pseudospectrum

### ASJC Scopus subject areas

- Software
- Theoretical Computer Science

### Cite this

*SIAM Journal on Optimization*,

*22*(2), 281-285. https://doi.org/10.1137/110822840

**Some regularity results for the pseudospectral abscissa and pseudospectral radius of a matrix.** / G̈urb̈uzbalaban, Mert; Overton, Michael L.

Research output: Contribution to journal › Article

*SIAM Journal on Optimization*, vol. 22, no. 2, pp. 281-285. https://doi.org/10.1137/110822840

}

TY - JOUR

T1 - Some regularity results for the pseudospectral abscissa and pseudospectral radius of a matrix

AU - G̈urb̈uzbalaban, Mert

AU - Overton, Michael L.

PY - 2012

Y1 - 2012

N2 - The ε-pseudospectral abscissa αε and radius ρε of an n×n matrix are, respectively, the maximal real part and the maximal modulus of points in its ε-pseudospectrum, defined using the spectral norm. It was proved in [A.S. Lewis and C.H.J. Pang, SIAM J. Optim., 19 (2008), pp. 1048-1072] that for fixed ε > 0, αε and ρε are Lipschitz continuous at a matrix A except when αε and ρε are attained at a critical point of the norm of the resolvent (in the nonsmooth sense), and it was conjectured that the points where αε and ρε are attained are not resolvent-critical. We prove this conjecture, which leads to the new result that αε and ρε are Lipschitz continuous, and also establishes the Aubin property with respect to both ε and A of the ε-pseudospectrum for the points z ε ℂ where αε and ρε are attained. Finally, we give a proof showing that the pseudospectrum can never be "pointed outwards."

AB - The ε-pseudospectral abscissa αε and radius ρε of an n×n matrix are, respectively, the maximal real part and the maximal modulus of points in its ε-pseudospectrum, defined using the spectral norm. It was proved in [A.S. Lewis and C.H.J. Pang, SIAM J. Optim., 19 (2008), pp. 1048-1072] that for fixed ε > 0, αε and ρε are Lipschitz continuous at a matrix A except when αε and ρε are attained at a critical point of the norm of the resolvent (in the nonsmooth sense), and it was conjectured that the points where αε and ρε are attained are not resolvent-critical. We prove this conjecture, which leads to the new result that αε and ρε are Lipschitz continuous, and also establishes the Aubin property with respect to both ε and A of the ε-pseudospectrum for the points z ε ℂ where αε and ρε are attained. Finally, we give a proof showing that the pseudospectrum can never be "pointed outwards."

KW - Aubin property

KW - Eigenvalue perturbation

KW - Lipschitz multifunction

KW - Pseudospectral abscissa

KW - Pseudospectral radius

KW - Pseudospectrum

UR - http://www.scopus.com/inward/record.url?scp=84865691297&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84865691297&partnerID=8YFLogxK

U2 - 10.1137/110822840

DO - 10.1137/110822840

M3 - Article

VL - 22

SP - 281

EP - 285

JO - SIAM Journal on Optimization

JF - SIAM Journal on Optimization

SN - 1052-6234

IS - 2

ER -