Geodesic
Condition numbers of Hessenberg companion matrices
Czechoslovak Mathematical Journal, Tome 74 (2024) no. 1, pp. 191-209
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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.\looseness -1
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.\looseness -1
DOI : 10.21136/CMJ.2024.0060-23
Classification : 15A12, 15B99
Keywords: companion matrix; Fiedler companion matrix; condition number; generalized companion matrix 
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Cox, Michael; Vander Meulen, Kevin N.; Van Tuyl, Adam; Voskamp, Joseph. Condition numbers of Hessenberg companion matrices. Czechoslovak Mathematical Journal, Tome 74 (2024) no. 1, pp. 191-209. doi: 10.21136/CMJ.2024.0060-23

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