Miniversal deformations of pairs of symmetric matrices under congruence

TL;DR

This paper introduces a minimal-parameter normal form, called a miniversal deformation, for pairs of complex symmetric matrices under congruence, facilitating smooth reductions of nearby matrix pairs.

Contribution

It provides explicit miniversal deformations for symmetric matrix pairs under congruence, including parameter counts and bounds on the distance to these normal forms.

Findings

Explicit miniversal normal forms for symmetric matrix pairs.

Calculation of the number of parameters in the deformation.

Upper bounds on the distance to the miniversal deformation.

Abstract

For each pair of complex symmetric matrices $(A,B)$ we provide a normal form with a minimal number of independent parameters, to which all pairs of complex symmetric matrices $(\widetilde{A},\widetilde{B})$, close to $(A,B)$ can be reduced by congruence transformation that smoothly depends on the entries of $\widetilde{A}$ and $\widetilde{B}$. Such a normal form is called a miniversal deformation of $(A,B)$ under congruence. A number of independent parameters in the miniversal deformation of a symmetric matrix pencil is equal to the codimension of the congruence orbit of this symmetric matrix pencil and is computed too. We also provide an upper bound on the distance from $(A,B)$ to its miniversal deformation.