Transformation to versal deformations of matrices

In the paper versal deformations of matrices are considered. The versal deformation is a matrix family inducing an arbitrary multi-parameter deformation of a given matrix by an appropriate smooth change of parameters and basis. Given a deformation of a matrix, it is suggested to find transformation functions (the change of parameters and the change of basis dependent on parameters) in the form of Taylor series. The general method of construction of recurrent procedures for calculation of coefficients in the Taylor expansions is developed and used for the cases of real and complex matrices, elements of classical Lie and Jordan algebras, and infinitesimally reversible matrices. Several examples are given and studied in detail. Applications of the suggested approach to problems of stability, singularity, and perturbation theories are discussed.