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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 congruence
Department of Computing Science and HPC2N, Umeå University, Umeå, Sweden.ORCID iD: 0000-0001-9110-6182
Department of Mathematics, University of São Paulo, São Paulo, Brazil.
Department of Computing Science and HPC2N, Umeå University, Umeå, Sweden.
National Technical University of Ukraine “Kyiv Polytechnic Institute”, Kiev, Ukraine.
2015 (English)In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 469, p. 305-334Article in journal (Refereed) Published
##### Abstract [en]

We construct the Hasse diagrams G2 and G3 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 + λAT 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 G2 B and G3 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.

##### Place, publisher, year, edition, pages
Elsevier, 2015. Vol. 469, p. 305-334
##### Keywords [en]
Bundle, Closure graph, Congruence canonical form, Congruence class, Perturbation
##### National Category
Mathematical Analysis
##### Identifiers
ISI: 000348883600014Scopus ID: 2-s2.0-84919935890OAI: oai:DiVA.org:oru-74885DiVA, id: diva2:1332901
##### Funder
eSSENCE - An eScience CollaborationSwedish Research Council, A0581501Available from: 2019-06-28 Created: 2019-06-28 Last updated: 2019-09-20Bibliographically approved

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#### Authority records BETA

Dmytryshyn, AndriiKågström, Bo

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Dmytryshyn, AndriiKågström, Bo
##### In the same journal
Linear Algebra and its Applications
##### On the subject
Mathematical Analysis

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Total: 59 hits

Cite
Citation style
• apa
• ieee
• modern-language-association-8th-edition
• vancouver
• Other style
More styles
Language
• de-DE
• en-GB
• en-US
• fi-FI
• nn-NO
• nn-NB
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• Other locale
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• text
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