Miniversal deformations of matrices under *congruence and reducing   transformations
  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]. 
A. Dmytryshyn, V. Futorny, V.V. Sergeichuk
math.RT