Groups and Symmetries in Numerical Linear Algebra
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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.
KeywordsAbelian Group Discrete Fourier Transform Short Exact Sequence Semidirect Product Schwartz Function
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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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