## How well-conditioned can the eigenvector problem be?

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Carlos Beltrán, Laurent Bétermin, Peter Grabner and Stefan Steinerberger
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## Abstract:

The condition number for eigenvector computations is a well-studied quantity. But how small can it possibly be? Specifically, what matrices are perfectly conditioned with respect to eigenvector computations? In this note we answer this question for $n \times n$ matrices, giving a solution that is exact to first-order as $n \rightarrow \infty$.## References

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## Additional Information

**Carlos Beltrán**- Affiliation: Facultad de Ciencias, Universidad of Cantabria, 39005 Santander, Spain
- MR Author ID: 764504
- ORCID: 0000-0002-0689-8232
- Email: beltranc@unican.es
**Laurent Bétermin**- Affiliation: Institut Camille Jordan, Université Claude Bernard Lyon 1, 69622 Villeurbanne, France
- ORCID: 0000-0003-4070-3344
- Email: betermin@math.univ-lyon1.fr
**Peter Grabner**- Affiliation: Institute of Analysis and Number Theory, Graz University of Technology, Kopernikusgasse 24, 8010 Graz, Austria
- MR Author ID: 293266
- ORCID: 0000-0002-0012-2302
- Email: peter.grabner@tugraz.at
**Stefan Steinerberger**- Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195
- MR Author ID: 869041
- ORCID: 0000-0002-7745-4217
- Email: steiner@uw.edu
- Received by editor(s): June 7, 2021
- Received by editor(s) in revised form: September 14, 2021
- Published electronically: December 7, 2021
- Additional Notes: The first author was supported by grant PID2020-113887GB-I00 funded by MCIN/ AEI /10.13039/501100011033; and by Banco de Santander and Universidad de Cantabria, through grant 21.SI01.64658. The second author was supported by the Austrian Science Fund (FWF) and the German Research Foundation (DFG) through the joint project FR 4083/3-1/I 4354 during his stay at the University of Vienna. The third author was supported by the Austrian Science Fund FWF project F5503 (part of the Special Research Program (SFB) “Quasi-Monte Carlo Methods: Theory and Applications”). The fourth author was partially supported by the NSF (DMS-2123224) and the Alfred P. Sloan Foundation (FG-2021-14114)
- © Copyright 2021 American Mathematical Society
- Journal: Math. Comp.
**91**(2022), 1237-1245 - MSC (2020): Primary 65F15; Secondary 31C20
- DOI: https://doi.org/10.1090/mcom/3706
- MathSciNet review: 4405494