@article {IOPORT.05268610,
    author       = {Burke, J.V. and Lewis, A.S. and Overton, M.L.},
    title        = {Convexity and Lipschitz behavior of small pseudospectra.},
    year         = {2008},
    journal      = {SIAM Journal on Matrix Analysis and Applications},
    volume       = {29},
    number       = {2},
    issn         = {0895-4798},
    pages        = {586-595},
    publisher    = {Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA},
    doi          = {10.1137/050645841},
    abstract     = {Consider the space $\Bbb{M}^{n}$ of all $n\times n$ complex matrices together with operator $2$-norm $\left\Vert ~\right\Vert $. We denote the spectrum of a matrix $A\in\Bbb{M}^{n}$ by $\Lambda(A).$ For each $\varepsilon>0$, the $\varepsilon$-pseudospectrum $\Lambda_{\varepsilon}(A)$ of $A$ is the union of the spectra $\Lambda(X)$ for all $X$ with $\left\Vert A-X\right\Vert \leq\varepsilon$. It is known that this pseudospectrum can be described as a finite union of sets each defined using finitely many polynomial inequalities; furthermore, each connected component of the pseudospectrum is compact and contains an eigenvalue of $A$, so, there are at most $n$ components [see {\it R. Benedetti} and {\it J.-J. Risler}, Real Algebraic and Semi-Algebraic Sets, Hermann, Paris (1990; Zbl 0694.14006)]. In general, a component of the pseudospectrum need not be convex. The main theorem of the present paper is the following. Suppose that $\lambda$ is a nonderogatory eigenvalue of a matrix $A_{0}$ in $\Bbb{M}^{n}$. For any sufficiently small $\mu>0$, there exists $\bar{\varepsilon}$ with $0<\bar{\varepsilon}<\mu$ such that for all $\varepsilon$ with $0<\varepsilon <\bar{\varepsilon}$ and all matrices $A$ in some neighbourhood of $A_{0}$ (depending on $\mu$ and $\varepsilon$), the set $$ \tilde{\Lambda}_{\varepsilon}(A):=\left\{ z\in\Lambda_{\varepsilon }(A)~:~\left\vert z-\lambda\right\vert <\mu\right\} $$ has the following properties: (i) $\tilde{\Lambda}_{\varepsilon}(A)$ is the component of $\Lambda_{\varepsilon}(A)$ containing $\lambda$ and it contains no other eigenvalue of $A_{0}$; (ii) $\tilde{\Lambda}_{\varepsilon}(A)$ is compact and strictly convex; (iii) the set value mapping $\tilde{\Lambda }_{\varepsilon}$ is Lipschitz on a neighbourhood of $A_{0}$ with respect to the Hausdorff distance. As a corollary it follows that, if $A_{0}$ is nonderogatory, then for all sufficiently small $\varepsilon>0$ the dependence of the pseudospectrum $\Lambda_{\varepsilon}(A)$ on the matrix $A$ is Lipschitz with respect to the Hausdorff measure for $A$ near $A_{0}$. In a note added in proof, the authors point out that this corollary has since been generalized in a recent preprint of {\it A.S. Lewis} and {\it C.H.J. Pang} [SIAM J. Optim. 19, No.3, 1048-1072 (2008)].},
    reviewer     = {John D. Dixon (Ottawa)},
    identifier   = {05268610},
}