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

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 $G_2^{\mathrm{B}}$ and $G_3^{\mathrm{B}}$ 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.

34 pages

References in corpus (10)

• Canonical forms for complex matrix congruence and *congruence
• Classification problems for system of forms and linear mappings
• Congruences of a square matrix and its transpose
• Canonical matrices of bilinear and sesquilinear forms
• A regularization algorithm for matrices of bilinear and sesquilinear forms
• Miniversal deformations of matrices of bilinear forms
• Miniversal deformations of matrices under *congruence and reducing transformations
• Simplest miniversal deformations of matrices, matrix pencils, and contragredient matrix pencils
• An informal introduction to perturbations of matrices determined up to similarity or congruence
• Tridiagonal canonical matrices of bilinear or sesquilinear forms and of pairs of symmetric, skew-symmetric, or Hermitian forms

Cited by in corpus (9)

• Miniversal deformations of matrices of bilinear forms
• Miniversal deformations of matrices under *congruence and reducing transformations
• An informal introduction to perturbations of matrices determined up to similarity or congruence
• On normal forms of complex points of small ${\mathcal{C}}^2$-perturbations of real $4$-manifolds embedded in a complex $3$-manifold
• Frobenius Degenerations of Preprojective Algebras
• Topological classification of systems of bilinear and sesquilinear forms
• The stabilizers for the action of orthogonal similarity on symmetric matrices and orthogonal $\ast$-conjugacy on Hermitian matrices
• Recovering a perturbation of a matrix polynomial from a perturbation of its linearization
• A constructive proof of Pokrzywa's theorem about perturbations of matrix pencils