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Finding eigenvalues of self-maps with the Kronecker canonical form. (English) Zbl 1390.15038
Kotsireas, Ilias S. (ed.) et al., Applications of computer algebra, Kalamata, Greece, July 20–23, 2015. Cham: Springer (ISBN 978-3-319-56930-7/hbk; 978-3-319-56932-1/ebook). Springer Proceedings in Mathematics & Statistics 198, 119-136 (2017).
Summary: 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.
For the entire collection see [Zbl 1379.13001].
MSC:
15A21 Canonical forms, reductions, classification
15A03 Vector spaces, linear dependence, rank, lineability
55N99 Homology and cohomology theories in algebraic topology
15A18 Eigenvalues, singular values, and eigenvectors
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References:
[1] 1. Edelsbrunner, H., Jabłoński, G., Mrozek, M.: The persistent homology of a self-map. Found. Comput. Math. 15 , 1213-1244 (2014) · Zbl 1330.55009
[2] 2. Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. Discrete Comput. Geom. 28 , 511-533 (2002) · Zbl 1011.68152
[3] 3. Gantmacher F.R.: The Theory of Matrices. Chelsea Publishing Company, New York (1959) · Zbl 0085.01001
[4] 4. Hartley, B., Hawkes, T.O.: Rings, Modules and Linear Algebra. Chapman and Hall, London (1970) · Zbl 0206.01603
[5] 5. van Dooren, P.: The computation of Kronecker’s canonical form of a singular pencil. Linear Algebra Appl. 27 , 103-140 (1979) · Zbl 0416.65026
[6] 6. Zomorodian, A., Carlsson, G.: Computing persistent homology. Discrete Comput. Geom. 33 , 249-274 (2005) · Zbl 1069.55003
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