About Hölder condition numbers and the stratification diagram for defective eigenvalues

Chaitin-Chatelin, Françoise; Harrabi, Abdellaziz; Ilahi, A.

doi:10.1016/s0378-4754(00)00178-6

Cited by 11 publications

(11 citation statements)

References 9 publications

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“…Furthermore, we show that for a nondegoratory eigenvalue of algebraic multiplicity m ≥ 2, 2) where N is the nilpotent operator associated with λ in the Jordan decomposition of A. The formulas (1.1) and (1.2) are the main results of this note.…”

mentioning

confidence: 85%

“…Now, we give the definition for the Hölder condition number of an eigenvalue of arbitrary multiplicity (see [2]). For λ ∈ C, m ∈ N and A ∈ C n×n we set If λ is an eigenvalue of A ∈ C n×n of algebraic multiplicity m then the Hölder condition number of λ to the order α > 0 is defined by…”

Section: Condition Numbersmentioning

confidence: 99%

“…Remark 4.4. In [7] (see also [2,4]) the following generalization of Theorem 4.3 has been shown. Let λ be an arbitrary eigenvalue of A.…”

mentioning

confidence: 94%

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Eigenvalue condition numbers and a formula of Burke, Lewis and Overton

Karow¹

2006

ELA

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mentioning

confidence: 85%

Section: Condition Numbersmentioning

confidence: 99%

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Eigenvalue condition numbers and a formula of Burke, Lewis and Overton

Karow¹

2006

ELA

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“…In this case however, we can still quantify conditioning by switching to a logarithmic scale [5,11,13]. Let us define the condition exponent of z by…”

Section: Lidskii's Perturbation Theorymentioning

confidence: 99%

Spectral conditioning and pseudospectral growth

2007

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Using the language of pseudospectra, we study the behavior of matrix eigenvalues under two scales of matrix perturbation. First, we relate Lidskii's analysis of small perturbations to a recent result of Karow on the growth rate of pseudospectra. Then, considering larger perturbations, we follow recent work of Alam and Bora in characterizing the distance from a given matrix to the set of matrices with multiple eigenvalues in terms of the number of connected components of pseudospectra.

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“…It measures the sensitivity of the eigenvalue λ if the matrix A is subjected to perturbations from the class Δ. In recent years some work has been done in order to obtain estimates or computable formulae for κ Δ (A, λ) [3,4,5,7,13,15,16,18,17,20,21,23]. However, the condition number cannot reveal how the eigenvalue moves in a specific direction under structured perturbations.…”

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confidence: 99%

Structured Pseudospectra and the Condition of a Nonderogatory Eigenvalue

Karow¹

2010

SIAM J. Matrix Anal. & Appl.

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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.

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About Hölder condition numbers and the stratification diagram for defective eigenvalues

Cited by 11 publications

References 9 publications

Eigenvalue condition numbers and a formula of Burke, Lewis and Overton

Eigenvalue condition numbers and a formula of Burke, Lewis and Overton

Spectral conditioning and pseudospectral growth

Structured Pseudospectra and the Condition of a Nonderogatory Eigenvalue

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