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Groups and Symmetries in Numerical Linear Algebra

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Exploiting Hidden Structure in Matrix Computations: Algorithms and Applications

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2173))

Abstract

Groups are fundamental objects of mathematics, describing symmetries of objects and also describing sets of motions moving points in a domain, such as translations in the plane and rotations of a sphere. The topic of these lecture notes is applications of group theory in computational mathematics. We will first cover fundamental properties of groups and continue with an extensive discussion of commutative (abelian) groups and their relationship to computational Fourier analysis. Various numerical algorithms will be discussed in the setting of group theory. Finally we will, more briefly, discuss generalisation of Fourier analysis to non-commutative groups and discuss problems in linear algebra with non-commutative symmetries. The representation theory of non-commutative finite groups is used as a tool to efficiently solve linear algebra problems with symmetries, exemplified by the computation of matrix exponentials.

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Acknowledgements

I would like to express a deep gratitude towards CIME and the organisers of this summer school for inviting me to present these lectures and for their patience with me during the tortuous process of writing the lecture notes. Also, I would like to thank Ulrich von der Ohe for his careful reading and commenting upon the manuscript.

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Authors and Affiliations

  1. Department of Mathematics, University of Bergen, 7803, N-5020, Bergen, Norway

    Hans Z. Munthe-Kaas

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  1. Hans Z. Munthe-Kaas
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Correspondence to Hans Z. Munthe-Kaas .

Editor information

Editors and Affiliations

  1. Mathematics and Science Center, Emory University, Atlanta, USA

    Michele Benzi

  2. Dipart. di Matematica, Università di Bologna, Bologna, Italy

    Valeria Simoncini

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Munthe-Kaas, H.Z. (2016). Groups and Symmetries in Numerical Linear Algebra. In: Benzi, M., Simoncini, V. (eds) Exploiting Hidden Structure in Matrix Computations: Algorithms and Applications . Lecture Notes in Mathematics(), vol 2173. Springer, Cham. https://doi.org/10.1007/978-3-319-49887-4_5

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