Name
Title | ||
|---|---|---|
Some Regularity Results for the Pseudospectral Abscissa and Pseudospectral Radius of a Matrix. |
Abstract | ||
|---|---|---|
The epsilon-pseudospectral abscissa alpha(epsilon) and radius rho(epsilon) of an n x n matrix are, respectively, the maximal real part and the maximal modulus of points in its epsilon-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 epsilon > 0, alpha(epsilon) and rho(epsilon) are Lipschitz continuous at a matrix A except when alpha(epsilon) and rho(epsilon) are attained at a critical point of the norm of the resolvent (in the nonsmooth sense), and it was conjectured that the points where alpha(epsilon) and rho(epsilon) are attained are not resolvent-critical. We prove this conjecture, which leads to the new result that alpha(epsilon) and rho(epsilon) are Lipschitz continuous, and also establishes the Aubin property with respect to both epsilon and A of the epsilon-pseudospectrum for the points z is an element of C where alpha(epsilon) and rho(epsilon) are attained. Finally, we give a proof showing that the pseudospectrum can never be "pointed outwards." |
Year | DOI | Venue |
|---|---|---|
2012 | 10.1137/110822840 | SIAM JOURNAL ON OPTIMIZATION |
Keywords | Field | DocType |
pseudospectrum,pseudospectral abscissa,pseudospectral radius,eigenvalue perturbation,Lipschitz multifunction,Aubin property | Discrete mathematics,Mathematical optimization,Pseudospectrum,Abscissa,Matrix (mathematics),Resolvent,Matrix norm,Critical point (thermodynamics),Lipschitz continuity,Conjecture,Mathematics | Journal |
Volume | Issue | ISSN |
22 | 2 | 1052-6234 |
Citations | PageRank | References |
2 | 0.42 | 2 |
Authors | ||
2 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Mert Gürbüzbalaban | 1 | 55 | 12.36 |
| Michael L. Overton | 2 | 634 | 590.15 |