Applied and Computational Mathematics

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A Matrix-Vector Construction of the Algebra of Complex Numbers

Received: 10 December 2018    Accepted: 02 January 2019    Published: 30 January 2019
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Abstract

The aim of this paper is to describe an alternative way to think about the algebra of complex numbers that may be of pedagogical value for introducing related concepts such as linear transformations and convolutions. The method is to define a fixed linear transformation of complex numbers represented in vector form so that products can be evaluated elementwise in the transformed space. The principal results are concrete demonstrations that this can in fact be accomplished.

DOI 10.11648/j.acm.20190801.11
Published in Applied and Computational Mathematics (Volume 8, Issue 1, February 2019)
Page(s) 1-2
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Algebras, Complex Numbers, Convolutions, Hypercomplex Algebras, Mathematics Education, Vector Spaces

References
[1] Lars Ahlfors, Complex Analysis (3rd ed.), McGraw-Hill, 1979.
[2] J. Hadamard, “Resolution d’une question relative aux determinants,” Bulletin des Sciences Mathematiques Series, 2 (17), pp. 240-246, 1893.
[3] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, 1990.
[4] W. Hamilton, ed., Elements of Quaternions, London (UK), 1866.
[5] I. L. Kantor and A. S. Solodvnikov, Hypercomplex Numbers: An Elementary Introduction to Algebras, New York: Springer-Verlag, 1989.
Author Information
  • Department of Electrical Engineering and Computer Science, University of Missouri, Columbia, USA

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    Jeffrey Uhlmann. (2019). A Matrix-Vector Construction of the Algebra of Complex Numbers. Applied and Computational Mathematics, 8(1), 1-2. https://doi.org/10.11648/j.acm.20190801.11

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    Jeffrey Uhlmann. A Matrix-Vector Construction of the Algebra of Complex Numbers. Appl. Comput. Math. 2019, 8(1), 1-2. doi: 10.11648/j.acm.20190801.11

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    Jeffrey Uhlmann. A Matrix-Vector Construction of the Algebra of Complex Numbers. Appl Comput Math. 2019;8(1):1-2. doi: 10.11648/j.acm.20190801.11

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      author = {Jeffrey Uhlmann},
      title = {A Matrix-Vector Construction of the Algebra of Complex Numbers},
      journal = {Applied and Computational Mathematics},
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      number = {1},
      pages = {1-2},
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      eprint = {https://download.sciencepg.com/pdf/10.11648.j.acm.20190801.11},
      abstract = {The aim of this paper is to describe an alternative way to think about the algebra of complex numbers that may be of pedagogical value for introducing related concepts such as linear transformations and convolutions. The method is to define a fixed linear transformation of complex numbers represented in vector form so that products can be evaluated elementwise in the transformed space. The principal results are concrete demonstrations that this can in fact be accomplished.},
     year = {2019}
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