| 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 |
| . |
| Category:
|
math |
| . |
| 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 |
| . |
| Date available:
|
2024-03-13T10:07:26Z |
| Last updated:
|
2026-04-06 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/152275 |
| . |
| 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 |
| . |
