Transformation of Families of Matrices to Normal Forms and its Application to Stability Theory

Abstract

Families of matrices smoothly depending on a vector of parameters are considered. Arnold [Russian Math. Surveys, 26 (1971), pp. 29--43] and Galin [Uspekhi Mat. Nauk, 27 (1972), pp. 241--242] have found and listed normal forms of families of complex and real matrices (miniversal deformations), to which any family of matrices can be transformed in the vicinity of a point in the parameter space by a change of basis, smoothly dependent on a vector of parameters, and by a smooth change of parameters. In this paper a constructive method of determining functions describing a change of basis and a change of parameters, transforming an arbitrary family to the miniversal deformation, is suggested. Derivatives of these functions with respect to parameters are determined from a recurrent procedure using derivatives of the functions of lower orders and derivatives of the family of matrices. Then the functions are found as Taylor series. Examples are given. The suggested method allows using efficiently miniversal deformations for investigation of different properties of matrix families. This is shown in the paper where tangent cones (linear approximations) to the stability domain at the singular boundary points are found.