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Change of the congruence canonical form of 2-by-2 and 3-by-3 matrices under perturbations and bundles of matrices under congruencePrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true}); PrimeFaces.cw("SelectBooleanButton","widget_formSmash_j_idt185",{id:"formSmash:j_idt185",widgetVar:"widget_formSmash_j_idt185",onLabel:"Hide others and affiliations",offLabel:"Show others and affiliations"});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 2015 (English)In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 469, 305-334 p.Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

2015. Vol. 469, 305-334 p.
##### Keyword [en]

Bundle, Closure graph, Congruence canonical form, Congruence class, Perturbation
##### National Category

Mathematics
##### Research subject

Mathematics; Computing Science
##### Identifiers

URN: urn:nbn:se:umu:diva-101053DOI: 10.1016/j.laa.2014.11.004ISI: 000348883600014ScopusID: 2-s2.0-84919935890OAI: oai:DiVA.org:umu-101053DiVA: diva2:796288
#####

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##### Funder

eSSENCE - An eScience CollaborationSwedish Research Council, A0581501
Available from: 2015-03-18 Created: 2015-03-18 Last updated: 2015-04-26Bibliographically approved

We construct the Hasse diagrams G_{2} and G_{3} for the closure ordering on the sets of congruence classes of 2 × 2 and 3 × 3 complex matrices. In other words, we construct two directed graphs whose vertices are 2 × 2 or, respectively, 3 × 3 canonical matrices under congruence, and there is a directed path from A to B if and only if A can be transformed by an arbitrarily small perturbation to a matrix that is congruent to B. A bundle of matrices under congruence is defined as a set of square matrices A for which the pencils A + λA^{T} belong to the same bundle under strict equivalence. In support of this definition, we show that all matrices in a congruence bundle of 2 × 2 or 3 × 3 matrices have the same properties with respect to perturbations. We construct the Hasse diagrams G_{2} ^{B} and G_{3} ^{B} for the closure ordering on the sets of congruence bundles of 2 × 2 and, respectively, 3 × 3 matrices. We find the isometry groups of 2 × 2 and 3 × 3 congruence canonical matrices.

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