### 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) |
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Pages (from-to) | 281-285 |

Number of pages | 5 |

Journal | SIAM Journal on Optimization |

Volume | 22 |

Issue number | 2 |

DOIs | |

Publication status | Published - Sep 7 2012 |

Externally published | Yes |

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### All Science Journal Classification (ASJC) codes

- Software
- Theoretical Computer Science

### Keywords

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

### Cite this

*SIAM Journal on Optimization*,

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