Transformation of matrix families to normal forms and its application to stability problems

Families of matrices, smoothly dependent on a vector of parameters, are considered. V. I. Arnold (1971) has found normal forms of families of complex matrices (miniversal deformations), such that any family of matrices in the vicinity of a point can be transformed to them by smoothly dependent on the vector of parameters change of basis and smooth change of parameters. Miniversal deformations of real matrices have been studied by D. M. Galin (1972). In this paper a method of determining functions describing change of basis and change of parameters, transforming arbitrary family to the miniversal deformation, is suggested. The functions are found as Taylor series, where derivatives of the functions are determined from a recurrent procedure using derivatives of these functions of lower orders and derivatives of the family. Examples are given. The results obtained allow to use miniversal deformations for investigation of different properties of matrix families more efficiently. This is shown in the paper, where tangent cones to the stability domain (linear approximations) at boundary points are found.