# Change of the congruence canonical form of 2-by-2 and 3-by-3 matrices under perturbations and bundles of matrices under congruence - Mathematics > Representation Theory

Change of the congruence canonical form of 2-by-2 and 3-by-3 matrices under perturbations and bundles of matrices under congruence - Mathematics > Representation Theory - Download this document for free, or read online. Document in PDF available to download.

Abstract: We construct the Hasse diagrams $G 2$ and $G 3$ for the closure ordering onthe sets of congruence classes of $2\times 2$ and $3\times 3$ complex matrices.In other words, we construct two directed graphs whose vertices are $2\times 2$or, respectively, $3\times 3$ canonical matrices under congruence and there isa directed path from $A$ to $B$ if and only if $A$ can be transformed by anarbitrarily 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 $A+\lambda A^T$ belong to the same bundle understrict equivalence. In support of this definition, we show that all matrices ina congruence bundle of $2\times 2$ or $3\times 3$ matrices have the sameproperties with respect to perturbations. We construct the Hasse diagrams$G 2^{ m B}$ and $G 3^{ m B}$ for the closure ordering on the sets ofcongruence bundles of $2\times 2$ and, respectively, $3\times 3$ matrices. Wefind the isometry groups of $2\times 2$ and $3\times 3$ congruence canonicalmatrices.

Author: ** Andrii Dmytryshyn, Vyacheslav Futorny, Bo Kågström, Lena Klimenko, Vladimir V. Sergeichuk**

Source: https://arxiv.org/