$D$-Stratification and Hierarchy Graphs of the Space of Order 2 and 3 Matrix Pencils

Abstract

Small changes in the entries of a matrix pencil may lead to important changes in its Kronecker normal form. Studies about the effect of small perturbations have been made when considering the stratification associated with the strict equivalence between matrix pencils. In this work, we consider a partition in the space of pairs of matrices associated to regular matrix pencils, which will be proved to be a finite stratification of the space of such matrix pencils, called D-stratification. Matrix pencils in the same strata are those having some prescribed Segre indices. We study the effect of perturbations which lead to changes in the Kronecker canonical form, preserving the order of the nilpotent part. Our goal is to determine which $D$-strata can be reached. In the cases where the order of the matrix pencils is 2 or 3, we obtain the corresponding hierarchy graphs, illustrating the $D$-strata that can be reached when applying some small perturbations.