Miniversal deformations of matrices of bilinear forms

TL;DR

This paper extends Arnold's concept of miniversal deformations from similarity to congruence, providing a simple normal form for matrices of bilinear forms that vary smoothly with small perturbations.

Contribution

It introduces a miniversal deformation framework for matrices under congruence, filling a gap in the theory of matrix perturbations for bilinear forms.

Findings

Constructed a miniversal deformation for matrices under congruence.

Provided a simple normal form for matrices of bilinear forms.

Extended Arnold's deformation theory to congruence transformations.

Abstract

V.I. Arnold [Russian Math. Surveys 26 (2) (1971) 29-43] constructed a miniversal deformation of matrices under similarity; that is, a simple normal form to which not only a given square matrix A but all matrices B close to it can be reduced by similarity transformations that smoothly depend on the entries of B. We construct a miniversal deformation of matrices under congruence.