Structured Pseudospectra and the Condition of a Nonderogatory Eigenvalue

Abstract

Abstract. Let λ be a nonderogatory eigenvalue of A ∈ C n×n of algebraic multiplicity m. The sensitivity of λ with respect to matrix perturbations of the form A A + Δ, Δ ∈ Δ, is measured by the structured condition number κ Δ (A, λ). Here Δ denotes the set of admissible perturbations. However, if Δ is not a vector space over C, then κ Δ (A, λ) provides only incomplete information about the mobility of λ under small perturbations from Δ. The full information is then given by the set K Δ (x, y) = {y * Δx; Δ ∈ Δ, Δ ≤ 1} ⊂ C that depends on Δ, a pair of normalized right and left eigenvectors x, y, and the norm · that measures the size of the perturbations. We always have κ Δ (A, λ) = max{|z| 1/m ; z ∈ K Δ (x, y)}. Furthermore, K Δ (x, y) determines the shape and growth of the Δ-structured pseudospectrum in a neighborhood of λ. In this paper we study the sets K Δ (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.

Structured Hölder Condition Numbers for Multiple Eigenvalues

“…Remark 3.9 Note that Theorem 3.8 (iv) also improves the results in [16,Theorem 4.3] and [32, Theorem 3.2], which only state bounds, but no explicit formula for the structured condition number of a simple nonzero eigenvalue. Recently, Karow [15] described the limit sets of the structured pseudospectra for complex skew-symmetric matrices, from which Theorem 3.8 (iv) could also be derived.…”

Structured Pseudospectra for Small Perturbations

“…The latter have been defined and investigated in [27]. For structured condition numbers of simple eigenvalues, see, e.g., [5,6,8,9,16,22,23,29,30,37,38]. Finally, we apply our results to the case of real perturbations of real matrices.…”

“…The sets (4.14) have been investigated in [23]. It has been shown there that these sets are ellipses in many important cases.…”

Computing unstructured and structured polynomial pseudospectrum approximations

“…The pseudospectrum is an important aid for shedding light on the sensitivity. Many properties and applications of the pseudospectrum of a matrix are discussed by Trefethen and Embree [23]; see also [6,7,10,13,20]. However, the computation of pseudospectra is a computationally demanding task except for very small matrices.…”