- SOME REGULARITY RESULTS FOR THE PSEUDOSPECTRAL ABSCISSA AND PSEUDOSPECTRAL RADIUS OF A MATRIX Mert Gurbuzbalaban(mertcims.nyu.edu) Michael L. Overton(overtoncs.nyu.edu) 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. Variational analysis of pseudospectra. SIAM Journal on Optimization, 19:1048-1072, 2008] 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 in the complex plane where $\alpha_\epsilon$ and $\rho_\epsilon$ are attained. Finally, we give a proof showing that the pseudospectrum can never be "pointed outwards". Keywords: Pseudospectrum, pseudospectral abscissa, pseudospectral radius,eigenvalue perturbation, Lipschitz multifunction, Aubin property Category 1: Convex and Nonsmooth Optimization (Nonsmooth Optimization ) Category 2: Global Optimization (Theory ) Category 3: Nonlinear Optimization Citation: Submitted. Jan 2011. Download: [PDF]Entry Submitted: 02/14/2011Entry Accepted: 02/14/2011Entry Last Modified: 02/14/2011Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Programming Society.