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

TL;DR

This paper constructs Hasse diagrams for the closure ordering of congruence classes and bundles of 2x2 and 3x3 matrices, analyzing their perturbation behavior and isometry groups.

Contribution

It introduces the construction of Hasse diagrams for congruence classes and bundles of small matrices, providing new insights into their perturbation structure and symmetries.

Findings

Constructed Hasse diagrams for congruence classes of 2x2 and 3x3 matrices.

Defined and analyzed congruence bundles, showing matrices in a bundle share perturbation properties.

Determined the isometry groups of canonical matrices for 2x2 and 3x3 matrices.

Abstract

We construct the Hasse diagrams $G_2$ and $G_3$ for the closure ordering on the 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 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 $A+\lambda A^T$ belong to the same bundle under strict equivalence. In support of this definition, we show that all matrices in a congruence bundle of $2\times 2$ or $3\times 3$ matrices have the same properties with respect to perturbations. We construct the Hasse diagrams $G_2^{\rm B}$ and $G_3^{\rm B}$ for the closure ordering on the sets of congruence bundles of $2\times 2$ and, respectively, $3\times 3$ matrices. We find the isometry groups of $2\times 2$ and $3\times 3$ congruence canonical matrices.