Change of the congruence canonical form of 2-by-2 and 3-by-3 matrices under perturbations and bundles of matrices under congruence
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
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.
Bundle, Closure graph, Congruence canonical form, Congruence class, Perturbation
Research subject Mathematics; Computing Science
IdentifiersURN: 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
FundereSSENCE - An eScience CollaborationSwedish Research Council, A0581501