Finding Eigenvalues of Self-maps with the Kronecker Canonical Form

  • Marc EthierEmail author
  • Grzegorz Jabłoński
  • Marian Mrozek
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 198)


Recent research has examined how to study the topological features of a continuous self-map by means of the persistence of the eigenspaces, for given eigenvalues, of the endomorphism induced in homology over a field. This raised the question of how to select dynamically significant eigenvalues. The present paper aims to answer this question, giving an algorithm that computes the persistence of eigenspaces for every eigenvalue simultaneously, also expressing said eigenspaces as direct sums of “finite” and “singular” subspaces.


Computational topology Persistent homology Self-maps Matrix pencils Kronecker canonical form 


  1. 1.
    Edelsbrunner, H., Jabłoński, G., Mrozek, M.: The persistent homology of a self-map. Found. Comput. Math. 15, 1213–1244 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. Discrete Comput. Geom. 28, 511–533 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Gantmacher F.R.: The Theory of Matrices. Chelsea Publishing Company, New York (1959)Google Scholar
  4. 4.
    Hartley, B., Hawkes, T.O.: Rings, Modules and Linear Algebra. Chapman and Hall, London (1970)zbMATHGoogle Scholar
  5. 5.
    van Dooren, P.: The computation of Kronecker’s canonical form of a singular pencil. Linear Algebra Appl. 27, 103–140 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Zomorodian, A., Carlsson, G.: Computing persistent homology. Discrete Comput. Geom. 33, 249–274 (2005)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Marc Ethier
    • 1
    • 2
    Email author
  • Grzegorz Jabłoński
    • 1
    • 3
  • Marian Mrozek
    • 1
  1. 1.Division of Computational Mathematics, Faculty of Mathematics and Computer ScienceJagiellonian UniversityKrakówPoland
  2. 2.University of Saint-BonifaceWinnipegCanada
  3. 3.Institute of Science and Technology AustriaKlosterneuburgAustria

Personalised recommendations