2010
DOI: 10.1137/070695836
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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, … Show more

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Cited by 13 publications
(8 citation statements)
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“…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.…”
Section: Symmetric Skew-symmetric and Hermitian Matricesmentioning
confidence: 99%
“…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.…”
Section: Symmetric Skew-symmetric and Hermitian Matricesmentioning
confidence: 99%
“…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.…”
Section: Introductionmentioning
confidence: 99%
“…The sets (4.14) have been investigated in [23]. It has been shown there that these sets are ellipses in many important cases.…”
Section: δ Is Invariant Under Complex Multiplication) Thenmentioning
confidence: 99%
“…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.…”
Section: Introductionmentioning
confidence: 99%