A holomorphic transformation to a miniversal deformation under *congruence does not always exist

TL;DR

This paper investigates the existence of holomorphic reduction transformations to miniversal deformations of matrices under *congruence, showing that such transformations do not always exist, unlike in similarity and congruence cases.

Contribution

It proves that holomorphic reduction transformations to miniversal deformations under *congruence are not always possible, highlighting a fundamental difference from similarity and congruence cases.

Findings

Holomorphic transformations exist for similarity and congruence cases.

Such transformations do not always exist for *congruence.

This reveals a limitation in the holomorphic reduction theory for matrices.

Abstract

V.I. Arnold [Russian Math. Surveys 26(2) (1971) 29-43] constructed miniversal deformations of square complex matrices under similarity. Reduction transformations to them and also to miniversal deformations of matrix pencils and matrices under congruence can be taken holomorphic. We prove that this is not true for reduction transformations to miniversal deformations of matrices under *congruence.