Change of the congruence canonical form of 2-by-2 and 3-by-3 matrices under perturbations and bundles of matrices under congruence

Autor(es)

Dmytryshyn, Andrii ^{[1, 2]} ; Futorny, Vyacheslav ^[3] ; Kagstrom, Bo ^{[1, 2]} ; Klimenko, Lena ^[4] ; Sergeichuk, Vladimir V. ^[5]

Afiliação do(s) autor(es)

^[1] Umea Univ, Dept Comp Sci, S-90187 Umea - Sweden

^[2] Umea Univ, HPC2N, S-90187 Umea - Sweden

^[3] Univ Sao Paulo, Dept Math, BR-05508 Sao Paulo - Brazil

^[4] Natl Tech Univ Ukraine, Kyiv Polytech Inst, Kiev - Ukraine

^[5] Ukrainian Acad Sci, Inst Math, Kiev - Ukraine

• Tipo de documento: Artigo Científico
• Fonte: Linear Algebra and its Applications; v. 469, p. 305-334, MAR 15 2015.
• Citações Web of Science: 5

Resumo

We construct the Hasse diagrams G(2) and G(3) for the closure ordering on the sets of congruence classes of 2 x 2 and 3 x 3 complex matrices. In other words, we construct two directed graphs whose vertices are 2 x 2 or, respectively, 3 x 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)AT belong to the same bundle under strict equivalence. In support of this definition, we show that all matrices in a congruence bundle of 2 x 2 or 3 x 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 x 2 and, respectively, 3 x 3 matrices. We find the isometry groups of 2 x 2 and 3 x 3 congruence canonical matrices. (C) 2014 Elsevier Inc. All rights reserved. (AU)

• Processo FAPESP: 10/50347-9 - Álgebras, representações e aplicações
• Beneficiário: Ivan Chestakov
• Modalidade de apoio: Auxílio à Pesquisa - Temático
• Processo FAPESP: 12/18139-2 - Métodos de teoria de representações em álgebra linear
• Beneficiário: Vyacheslav Futorny
• Modalidade de apoio: Auxílio à Pesquisa - Pesquisador Visitante - Internacional