Miniversal deformations of matrices under *congruence and reducing transformations

TL;DR

This paper extends Arnold's concept of miniversal deformations from similarity to *congruence and congruence of matrices, providing explicit constructions and analytic transformations for these cases.

Contribution

It introduces miniversal deformations of matrices under *congruence and constructs analytic reducing transformations, expanding the theory beyond similarity transformations.

Findings

Constructed miniversal deformations under *congruence.

Provided explicit analytic reducing transformations.

Extended previous results on congruence to *congruence.

Abstract

V.I. Arnold [Russian Math. Surveys 26(2) (1971) 29-43] constructed a miniversal deformation of a square complex matrix 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 give miniversal deformations of matrices of sesquilinear forms; that is, of square complex matrices under *congruence, and construct an analytic reducing transformation to a miniversal deformation. Analogous results for matrices under congruence were obtained by the authors in [Linear Algebra Appl. 436 (2012) 2670-2700].