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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
Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
University of Sao Paulo, Brazil .
Umeå University, Faculty of Science and Technology, Department of Computing Science. Umeå University, Faculty of Science and Technology, High Performance Computing Center North (HPC2N).
Kiev Polytechnic Institute, Ukraine.
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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]

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
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: 000348883600014Scopus ID: 2-s2.0-84919935890OAI: oai:DiVA.org:umu-101053DiVA: diva2:796288
Funder
eSSENCE - An eScience CollaborationSwedish Research Council, A0581501
Available from: 2015-03-18 Created: 2015-03-18 Last updated: 2017-12-04Bibliographically approved

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Dmytryshyn, AndriiKågström, Bo

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Dmytryshyn, AndriiKågström, Bo
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Department of Computing ScienceHigh Performance Computing Center North (HPC2N)
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Linear Algebra and its Applications
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