Computation of multiple eigenvalues and generalized eigenvectors for matrices dependent on parameters

The paper develops Newton's method of finding multiple eigenvalues with one Jordan block and corresponding generalized eigenvectors for matrices dependent on parameters. It computes the nearest value of a parameter vector with a matrix having a multiple eigenvalue of given multiplicity. The method also works in the whole matrix space (in the absence of parameters). The approach is based on the versal deformation theory for matrices. Numerical examples are given. The implementation of the method in MATLAB code is available.Comment: 19 pages, 3 figure

D-stratification and hierarchy graphs of the space of order 2 and 3 matrix pencils

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 DD-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 DD-strata that can be reached when applying some small perturbationsPeer ReviewedPostprint (author's final draft

Computation and Presentation of Graphs Displaying Closure Hierarchies of Jordan and Kronecker Structures

StratiGraph, a Java-based tool for computation and presentation of closure hierarchies of Jordan and Kronecker structures is presented. The too lis based on recent theoretical results on strati cations of orbits and bundles of matrices and matrix pencils. A stratification reveals the complete hierarchy of nearby structures, information critical for explaining the qualitative behaviour of linear systems under perturbations. StratiGraph facilitates the application of these theories and visualizes the resulting hierarchy as a graph. Nodes in the graph represent orbits or bundles of matrices or matrix pencils. Edges represent covering relations in the closure hierarchy. Given a Jordan or Kronecker structure, a user can obtain the complete information of nearby structures simply by mouse clicks on nodes of interest. This contribution gives an overview of the StratiGraph tool, presents its main functionalities and other features, and illustrates its use by sample applications