Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2018
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/1901.00050 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909434351976448 |
|---|---|
| author | Lewis, Adrian S. Overton, Michael L. |
| author_facet | Lewis, Adrian S. Overton, Michael L. |
| contents | Solutions to optimization problems involving the numerical radius often belong to a special class: the set of matrices having field of values a disk centered at the origin. After illustrating this phenomenon with some examples, we illuminate it by studying matrices around which this set of "disk matrices" is a manifold with respect to which the numerical radius is partly smooth. We then apply our results to matrices whose nonzeros consist of a single superdiagonal, such as Jordan blocks and the Crabb matrix related to a well-known conjecture of Crouzeix. Finally, we consider arbitrary complex three-by-three matrices; even in this case, the details are surprisingly intricate. One of our results is that in this real vector space with dimension 18, the set of disk matrices is a semi-algebraic manifold with dimension 12. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1901_00050 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Partial smoothness of the numerical radius at matrices whose fields of values are disks Lewis, Adrian S. Overton, Michael L. Numerical Analysis Optimization and Control Solutions to optimization problems involving the numerical radius often belong to a special class: the set of matrices having field of values a disk centered at the origin. After illustrating this phenomenon with some examples, we illuminate it by studying matrices around which this set of "disk matrices" is a manifold with respect to which the numerical radius is partly smooth. We then apply our results to matrices whose nonzeros consist of a single superdiagonal, such as Jordan blocks and the Crabb matrix related to a well-known conjecture of Crouzeix. Finally, we consider arbitrary complex three-by-three matrices; even in this case, the details are surprisingly intricate. One of our results is that in this real vector space with dimension 18, the set of disk matrices is a semi-algebraic manifold with dimension 12. |
| title | Partial smoothness of the numerical radius at matrices whose fields of values are disks |
| topic | Numerical Analysis Optimization and Control |
| url | https://arxiv.org/abs/1901.00050 |