Eigenvalue condition numbers: Zero-structured versus traditional

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.

“…The following theorem extends a result by Noschese and Pasquini [17] on zerostructured perturbations.…”

Section: Corollary 42 Let 1 ≤ R P ≤ ∞ and Let T Q Be Given By (4supporting

confidence: 82%

mentioning

confidence: 77%

mentioning

confidence: 99%

See 1 more Smart Citation

Structured Pseudospectra and the Condition of a Nonderogatory Eigenvalue

Karow¹

2010

“…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: 92%

Structured Pseudospectra for Small Perturbations

Karow¹

2011

“…Noschese and Pasquini [16] show that for perturbations having an assigned zero structure (or sparsity pattern), (2.14) reduces to…”

Section: Structured Condition Numbermentioning

confidence: 99%

Structured Eigenvalue Condition Numbers

Karow

Kreßner²,

Tisseur

2006

Abstract. This paper investigates the effect of structure-preserving perturbations on the eigenvalues of linearly and nonlinearly structured eigenvalue problems. Particular attention is paid to structures that form Jordan algebras, Lie algebras, and automorphism groups of a scalar product. Bounds and computable expressions for structured eigenvalue condition numbers are derived for these classes of matrices, which include complex symmetric, pseudo-symmetric, persymmetric, skewsymmetric, Hamiltonian, symplectic, and orthogonal matrices. In particular we show that under reasonable assumptions on the scalar product, the structured and unstructured eigenvalue condition numbers are equal for structures in Jordan algebras. For Lie algebras, the effect on the condition number of incorporating structure varies greatly with the structure. We identify Lie algebras for which structure does not affect the eigenvalue condition number.