Perturbation of matrices preserving the Segre characteristics of the non-nilpotent Jordan blocks

Abstract

Abstract We consider in the space of square matrices with complex co- efficients the following equivalence relation: two matrices are said to be equivalent if and only if the direct sum of the non- nilpotent Jordan blocks in their canonical Jordan form are the same (up to permutation of the blocks), and allowing the nilpo- tent Jordan blocks to be different. This condition is equivalent to having the same Drazin inverses. We study the structure of the sets consisting of matrices which satisfy this condition (that is to say, equivalence classes). If we consider the sets of matrices which have the same Segre characteristics for the non-zero eigenvalues, allowing their val- ues to be different, we obtain a stratification. We also investi- gate the effect of small perturbations, constructing closure hi- erarchies and identifying more generic and more degenerated nearby matrices to a given one.