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Spectral Properties and Optimality for Elementary Matrices

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Abstract

A generalization of the Householder transformation, renamed as elementary matrix by A.S. Householder: Unitary transformation of a nonsymmetric matrix, J. ACM, 5(4), 339–342, 1958, was introduced by LaBudde (Math Comput 17(84):433–437, 1963) as a tool to obtain a tridiagonal matrix similar to a given square matrix. Some of the free parameters of the transformation can be chosen to attain better numerical properties. In this work, we study the spectral properties of the transformation. We also propose a special choice for free coefficients of that transformation to minimize its condition number. The transformation with such suitable choice of parameters is called optimal.

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Acknowledgements

The authors thank Prof. F. Bazán for his helpful comments and references [7, 8]. The first author thanks the support of the sponsors of the Wave Inversion Technology Consortium.

The authors are in debit with Gene Golub for his valuable contributions to this paper, specially for Theorem 2.2.

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Authors and Affiliations

  1. Department of Applied Mathematics, University of Campinas & INCT-GP, Campinas, 13083-859, Brazil

    Ricardo Biloti

  2. Instituto Kumon de Educação Ltda., São Paulo, 04006-002, Brazil

    João Daniel Palma Ramos

  3. Department of Mathematics, Federal University of Paraná, Curitiba, 81531-980, Brazil

    Jin-Yun Yuan

Authors
  1. Ricardo Biloti
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  2. João Daniel Palma Ramos
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  3. Jin-Yun Yuan
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Corresponding author

Correspondence to Ricardo Biloti.

Additional information

The work of the first and third authors was partially supported by National Council for Scientific and Technological Development (CNPq), Brazil.

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Biloti, R., Ramos, J.D.P. & Yuan, JY. Spectral Properties and Optimality for Elementary Matrices. J. Oper. Res. Soc. China 6, 467–472 (2018). https://doi.org/10.1007/s40305-017-0177-z

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  • DOI: https://doi.org/10.1007/s40305-017-0177-z

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