Structured pseudospectra and the condition of a nonderogatory eigenvalue
| dc.contributor.author | Karow, Michael | |
| dc.date.accessioned | 2017-12-14T15:20:48Z | |
| dc.date.available | 2017-12-14T15:20:48Z | |
| dc.date.issued | 2010-11-30 | |
| dc.description.abstract | Let $\lambda$ be a nonderogatory eigenvalue of $A\in\mathbb{C}^{n\times n}$ of algebraic multiplicity m. The sensitivity of $\lambda$ with respect to matrix perturbations of the form $A\leadsto A+\Delta$, $\Delta\in\boldsymbol{\Delta}$, is measured by the structured condition number $\kappa_{\boldsymbol{\Delta}}(A,\lambda)$. Here $\boldsymbol{\Delta}$ denotes the set of admissible perturbations. However, if $\boldsymbol{\Delta}$ is not a vector space over $\mathbb{C}$, then $\kappa_{\boldsymbol{\Delta}}(A,\lambda)$ provides only incomplete information about the mobility of $\lambda$ under small perturbations from $\boldsymbol{\Delta}$. The full information is then given by the set $K_{\boldsymbol{\Delta}}(x,y)=\{y^*\Delta x;$ $\Delta\in\boldsymbol{\Delta},$ $\|\Delta\|\leq1\}\subset\mathbb{C}$ that depends on $\boldsymbol{\Delta}$, a pair of normalized right and left eigenvectors $x,y$, and the norm $\|\cdot\|$ that measures the size of the perturbations. We always have $\kappa_{\boldsymbol{\Delta}}(A,\lambda)=\max\{|z|^{1/m};$ $z\in K_{\boldsymbol{\Delta}}(x,y)\}$. Furthermore, $K_{\boldsymbol{\Delta}}(x,y)$ determines the shape and growth of the $\boldsymbol{\Delta}$-structured pseudospectrum in a neighborhood of $\lambda$. In this paper we study the sets $K_{\boldsymbol{\Delta}}(x,y)$ 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. | en |
| dc.identifier.eissn | 1095-7162 | |
| dc.identifier.issn | 0895-4798 | |
| dc.identifier.uri | https://depositonce.tu-berlin.de/handle/11303/7272 | |
| dc.identifier.uri | http://dx.doi.org/10.14279/depositonce-6545 | |
| dc.language.iso | en | en |
| dc.rights.uri | http://rightsstatements.org/vocab/InC/1.0/ | en |
| dc.subject.ddc | 518 Numerische Analysis | de |
| dc.subject.ddc | 512 Algebra | de |
| dc.subject.other | eigenvalues | en |
| dc.subject.other | structured perturbations | en |
| dc.subject.other | pseudospectra | en |
| dc.subject.other | condition numbers | en |
| dc.title | Structured pseudospectra and the condition of a nonderogatory eigenvalue | en |
| dc.type | Article | en |
| dc.type.version | publishedVersion | en |
| dcterms.bibliographicCitation.doi | 10.1137/070695836 | en |
| dcterms.bibliographicCitation.issue | 5 | en |
| dcterms.bibliographicCitation.journaltitle | SIAM Journal on Matrix Analysis and Applications | en |
| dcterms.bibliographicCitation.originalpublishername | Society for Industrial and Applied Mathematics | en |
| dcterms.bibliographicCitation.originalpublisherplace | Philadelphia, Pa. | en |
| dcterms.bibliographicCitation.pageend | 2881 | en |
| dcterms.bibliographicCitation.pagestart | 2860 | en |
| dcterms.bibliographicCitation.volume | 31 | en |
| tub.accessrights.dnb | free | en |
| tub.affiliation | Fak. 2 Mathematik und Naturwissenschaften::Inst. Mathematik::FG Numerische Lineare Algebra | de |
| tub.affiliation.faculty | Fak. 2 Mathematik und Naturwissenschaften | de |
| tub.affiliation.group | FG Numerische Lineare Algebra | de |
| tub.affiliation.institute | Inst. Mathematik | de |
| tub.publisher.universityorinstitution | Technische Universität Berlin | en |