Let lambda be a nonderogatory eigenvalue of A is an element of C-nxn of algebraic multiplicity m. The sensitivity of lambda with respect to matrix perturbations of the form A (sic) A + Delta, Delta is an element of Delta, is measured by the structured condition number kappa(Delta)(A, lambda). Here Delta denotes the set of admissible perturbations. However, if Delta is not a vector space over C, then kappa(Delta)(A, lambda) provides only incomplete information about the mobility of lambda under small perturbations from Delta. The full information is then given by the set K-Delta( x, y) = {y*Delta x; Delta is an element of Delta, parallel to Delta parallel to <= 1} subset of C that depends on Delta, a pair of normalized right and left eigenvectors x, y, and the norm parallel to.parallel to that measures the size of the perturbations. We always have kappa(Delta)(A, lambda) = max{vertical bar z vertical bar(1/m); z is an element of K-Delta(x, y)}. Furthermore, K-Delta(x, y) determines the shape and growth of the Delta-structured pseudospectrum in a neighborhood of lambda. In this paper we study the sets K-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.
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