Convexity and Lipschitz Behavior of Small Pseudospectra
James V. Burke, Adrian S. Lewis, Michael L. Overton · SIAM Journal on Matrix Analysis and Applications · 2007
The ε‐pseudospectrum of a matrix A is the subset of the complex plane consisting of all eigenvalues of complex matrices within a distance ε of A, measured by the operator 2‐norm. Given a nonderogatory matrix $A_0$, for small $\epsilon > 0, we show that the ε‐pseudospectrum of any matrix A near $A_0$ consists of compact convex neighborhoods of the eigenvalues of $A_0$. Furthermore, the dependence of each of these neighborhoods on A is Lipschitz.