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Computational Aspects of Some Exponential Identities

Conference paper
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Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 267)

Abstract

The notion of the exponential of a matrix is usually introduced in elementary textbooks on ordinary differential equations when solving a constant coefficients linear system, also providing some of its properties and in particular one that does not hold unless the involved matrices commute. Several problems arise indeed from this fundamental issue, and it is our purpose to review some of them in this work, namely: (i) is it possible to write the product of two exponential matrices as the exponential of a matrix? (ii) is it possible to “disentangle” the exponential of a sum of two matrices? (iii) how to write the solution of a time-dependent linear differential system as the exponential of a matrix? To address these problems the Baker–Campbell–Hausdorff series, the Zassenhaus formula and the Magnus expansion are formulated and efficiently computed, paying attention to their convergence. Finally, several applications are also considered.

Keywords

Matrix exponential Baker–Campbell–Hausdorff series Zassenhaus formula Magnus expansion 

MSC codes

5A16 22E60 17B66 34L99 

Notes

Acknowledgements

The author would like to thank the three referees for their insightful remarks that have helped him to improve the paper. This work has been partially supported by Ministerio de Economía y Competitividad (Spain) through the coordinated project MTM2013-46553-C3 (co-funded by FEDER).

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Authors and Affiliations

  1. 1.Institut de Matemàtiques i Aplicacions de Castelló and Departament de MatemàtiquesUniversitat Jaume ICastellónSpain

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