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Condition numbers of Hessenberg companion matrices

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Abstract

The Fiedler matrices are a large class of companion matrices that include the well-known Frobenius companion matrix. The Fiedler matrices are part of a larger class of companion matrices that can be characterized by a Hessenberg form. We demonstrate that the Hessenberg form of the Fiedler companion matrices provides a straight-forward way to compare the condition numbers of these matrices. We also show that there are other companion matrices which can provide a much smaller condition number than any Fiedler companion matrix. We finish by exploring the condition number of a class of matrices obtained from perturbing a Frobenius companion matrix while preserving the characteristic polynomial.

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Authors and Affiliations

  1. Unit 202 - 133 Herkimer Street, Hamilton, Ontario, L8P 2H3, Canada

    Michael Cox

  2. Department of Mathematics, Redeemer University, 777 Garner Road East, Ancaster, Ontario, L9K 1J4, Canada

    Kevin N. Vander Meulen

  3. Department of Mathematics and Statistics, McMaster University, 1280 Main Street West, Hamilton Hall, Hamilton, Ontario, L8S 4L8, Canada

    Adam Van Tuyl & Joseph Voskamp

Authors
  1. Michael Cox
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  2. Kevin N. Vander Meulen
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  3. Adam Van Tuyl
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  4. Joseph Voskamp
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Corresponding author

Correspondence to Kevin N. Vander Meulen.

Additional information

Research of Vander Meulen was supported in part by NSERC Discovery Grant No. 2022-05137. Research of Van Tuyl was supported in part by NSERC Discovery Grant No. 2019-05412. Research of Voskamp was supported in part by NSERC USRA 504279.

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Cox, M., Vander Meulen, K.N., Van Tuyl, A. et al. Condition numbers of Hessenberg companion matrices. Czech Math J 74, 191–209 (2024). https://doi.org/10.21136/CMJ.2024.0060-23

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  • DOI: https://doi.org/10.21136/CMJ.2024.0060-23

Keywords

MSC 2020

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