Structured Pseudospectra and the Condition of a Nonderogatory Eigenvalue

Let λ be a nonderogatory eigenvalue of A ∈ C. The sensitivity of λ with respect to matrix perturbations A A + ∆,∆ ∈ ∆, is measured by the structured condition number κ∆(A,λ). Here ∆ denotes the set of admissible perturbations. However, if ∆ is not a vector space over C then κ∆(A, λ) provides only incomplete information about the mobility of λ under small perturbations from ∆. The full information is then given by a certain set K∆(x, y) ⊂ C which depends on ∆ and a pair of normalized right and left eigenvectors x, y. In this paper we study the sets K∆(x, y) and obtain methods for computing them. In particular we show that K∆(x, y) is an ellipse in some important cases.