On the computation of a versal family of matrices

Abstract

The aim of this article is to present algorithms for the computation of versal deformations of matrices. A deformation of a matrixA 0 is a holomorphic matrix-valued function whose value at a pointt 0∈C p is the matrixA 0. We want to study the properties of these matrices in a neighbourhood oft 0. One could, for each valuet in this neighbourhood, compute the Jordan form as well as the change of basis matrix; but, generally, the results will not be analytic. So, we want to construct a deformation of the matrixA 0 into which any deformation can be transformed by an invertible deformation of the matrixId. After having introduced the notion of versal deformation, we shall provide computer algebra algorithms to computer these normal forms. In the last section, we shall show that a one-parameter deformation can be transformed into a simpler form than the general versal deformation.