Czechoslovak Mathematical Journal, Vol. 74, No. 1, pp. 191-209, 2024
Condition numbers of Hessenberg companion matrices
Michael Cox, Kevin N. Vander Meulen, Adam Van Tuyl, Joseph Voskamp
Received February 8, 2023. Published online January 26, 2024.
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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Affiliations: Michael Cox, Unit 202 - 133 Herkimer Street, Hamilton, Ontario, L8P 2H3, Canada, e-mail: michael.m.cox@outlook.com; Kevin N. Vander Meulen (corresponding author), Department of Mathematics, Redeemer University, 777 Garner Road East, Ancaster, Ontario, L9K 1J4, Canada, e-mail: kvanderm@redeemer.ca; Adam Van Tuyl, Joseph Voskamp, Department of Mathematics and Statistics, McMaster University, 1280 Main Street West, Hamilton Hall, Ontario, L8S 4L8, Canada, e-mail: vantuyl@math.mcmaster.ca, voskampj@mcmaster.ca
