Article
| Title: | Condition numbers of Hessenberg companion matrices (English) |
| Author: | Cox, Michael |
| Author: | Vander Meulen, Kevin N. |
| Author: | Van Tuyl, Adam |
| Author: | Voskamp, Joseph |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 74 |
| Issue: | 1 |
| Year: | 2024 |
| Pages: | 191-209 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | 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 (English) |
| Keyword: | companion matrix |
| Keyword: | Fiedler companion matrix |
| Keyword: | condition number |
| Keyword: | generalized companion matrix |
| MSC: | 15A12 |
| MSC: | 15B99 |
| idZBL: | Zbl 07893374 |
| idMR: | MR4717829 |
| DOI: | 10.21136/CMJ.2024.0060-23 |
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| Date available: | 2024-03-13T10:07:26Z |
| Last updated: | 2024-12-13 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/152275 |
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| Reference: | [1] Cox, M.: On Conditions Numbers of Companion Matrices: M.Sc. Thesis.McMaster University, Hamilton (2018). |
| Reference: | [2] Deaett, L., Fischer, J., Garnett, C., Meulen, K. N. Vander: Non-sparse companion matrices.Electron. J. Linear Algebra 35 (2019), 223-247. Zbl 1419.15030, MR 3982283, 10.13001/1081-3810.3839 |
| Reference: | [3] Terán, F. de, Dopico, F. M., Pérez, J.: Condition numbers for inversion of Fiedler companion matrices.Linear Algebra Appl. 439 (2013), 944-981. Zbl 1281.15004, MR 3061748, 10.1016/j.laa.2012.09.020 |
| Reference: | [4] Eastman, B., Kim, I.-J., Shader, B. L., Meulen, K. N. Vander: Companion matrix patterns.Linear Algebra Appl. 463 (2014), 255-272. Zbl 1310.15015, MR 3262399, 10.1016/j.laa.2014.09.010 |
| Reference: | [5] Fiedler, M.: A note on companion matrices.Linear Algebra Appl. 372 (2003), 325-331. Zbl 1031.15014, MR 1999154, 10.1016/S0024-3795(03)00548-2 |
| Reference: | [6] Garnett, C., Shader, B. L., Shader, C. L., Driessche, P. van den: Characterization of a family of generalized companion matrices.Linear Algebra Appl. 498 (2016), 360-365. Zbl 1371.15019, MR 3478567, 10.1016/j.laa.2015.07.031 |
| Reference: | [7] Meulen, K. N. Vander, Vanderwoerd, T.: Bounds on polynomial roots using intercyclic companion matrices.Linear Algebra Appl. 539 (2018), 94-116. Zbl 1380.15011, MR 3739399, 10.1016/j.laa.2017.11.002 |
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