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Structured pseudospectra and the condition of a nonderogatory eigenvalue

Karow, Michael

FG Numerische Lineare Algebra

Let $\lambda$ be a nonderogatory eigenvalue of $A\in\mathbb{C}^{n\times n}$ of algebraic multiplicity m. The sensitivity of $\lambda$ with respect to matrix perturbations of the form $A\leadsto A+\Delta$, $\Delta\in\boldsymbol{\Delta}$, is measured by the structured condition number $\kappa_{\boldsymbol{\Delta}}(A,\lambda)$. Here $\boldsymbol{\Delta}$ denotes the set of admissible perturbations. However, if $\boldsymbol{\Delta}$ is not a vector space over $\mathbb{C}$, then $\kappa_{\boldsymbol{\Delta}}(A,\lambda)$ provides only incomplete information about the mobility of $\lambda$ under small perturbations from $\boldsymbol{\Delta}$. The full information is then given by the set $K_{\boldsymbol{\Delta}}(x,y)=\{y^*\Delta x;$ $\Delta\in\boldsymbol{\Delta},$ $\|\Delta\|\leq1\}\subset\mathbb{C}$ that depends on $\boldsymbol{\Delta}$, a pair of normalized right and left eigenvectors $x,y$, and the norm $\|\cdot\|$ that measures the size of the perturbations. We always have $\kappa_{\boldsymbol{\Delta}}(A,\lambda)=\max\{|z|^{1/m};$ $z\in K_{\boldsymbol{\Delta}}(x,y)\}$. Furthermore, $K_{\boldsymbol{\Delta}}(x,y)$ determines the shape and growth of the $\boldsymbol{\Delta}$-structured pseudospectrum in a neighborhood of $\lambda$. In this paper we study the sets $K_{\boldsymbol{\Delta}}(x,y)$ and obtain methods for computing them. In doing so we obtain explicit formulae for structured eigenvalue condition numbers with respect to many important perturbation classes.