Eigenvalue Attraction
We prove that the complex conjugate (c.c.) eigenvalues of a smoothly varying real matrix attract (Eq. 15). We offer a dynamical perspective on the motion and interaction of the eigenvalues in the complex plane, derive their governing equations and discuss applications. C.c. pairs closest to the real...
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| Format: | Article |
| Language: | English |
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Springer US
2017
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| Online Access: | http://hdl.handle.net/1721.1/106582 https://orcid.org/0000-0002-4078-6752 |
| _version_ | 1826202286451326976 |
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| author | Movassagh, Ramis |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Movassagh, Ramis |
| author_sort | Movassagh, Ramis |
| collection | MIT |
| description | We prove that the complex conjugate (c.c.) eigenvalues of a smoothly varying real matrix attract (Eq. 15). We offer a dynamical perspective on the motion and interaction of the eigenvalues in the complex plane, derive their governing equations and discuss applications. C.c. pairs closest to the real axis, or those that are ill-conditioned, attract most strongly and can collide to become exactly real. As an application we consider random perturbations of a fixed matrix M. If M is Normal, the total expected force on any eigenvalue is shown to be only the attraction of its c.c. (Eq. 24) and when M is circulant the strength of interaction can be related to the power spectrum of white noise. We extend this by calculating the expected force (Eq. 41) for real stochastic processes with zero-mean and independent intervals. To quantify the dominance of the c.c. attraction, we calculate the variance of other forces. We apply the results to the Hatano-Nelson model and provide other numerical illustrations. It is our hope that the simple dynamical perspective herein might help better understanding of the aggregation and low density of the eigenvalues of real random matrices on and near the real line respectively. In the appendix we provide a Matlab code for plotting the trajectories of the eigenvalues. |
| first_indexed | 2024-09-23T12:05:07Z |
| format | Article |
| id | mit-1721.1/106582 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T12:05:07Z |
| publishDate | 2017 |
| publisher | Springer US |
| record_format | dspace |
| spelling | mit-1721.1/1065822022-10-01T08:03:43Z Eigenvalue Attraction Movassagh, Ramis Massachusetts Institute of Technology. Department of Mathematics Movassagh, Ramis We prove that the complex conjugate (c.c.) eigenvalues of a smoothly varying real matrix attract (Eq. 15). We offer a dynamical perspective on the motion and interaction of the eigenvalues in the complex plane, derive their governing equations and discuss applications. C.c. pairs closest to the real axis, or those that are ill-conditioned, attract most strongly and can collide to become exactly real. As an application we consider random perturbations of a fixed matrix M. If M is Normal, the total expected force on any eigenvalue is shown to be only the attraction of its c.c. (Eq. 24) and when M is circulant the strength of interaction can be related to the power spectrum of white noise. We extend this by calculating the expected force (Eq. 41) for real stochastic processes with zero-mean and independent intervals. To quantify the dominance of the c.c. attraction, we calculate the variance of other forces. We apply the results to the Hatano-Nelson model and provide other numerical illustrations. It is our hope that the simple dynamical perspective herein might help better understanding of the aggregation and low density of the eigenvalues of real random matrices on and near the real line respectively. In the appendix we provide a Matlab code for plotting the trajectories of the eigenvalues. National Science Foundation (U.S.) (Grant DMS. 1312831) 2017-01-20T23:51:57Z 2017-01-20T23:51:57Z 2015-12 2015-11 2016-08-18T15:44:47Z Article http://purl.org/eprint/type/JournalArticle 0022-4715 1572-9613 http://hdl.handle.net/1721.1/106582 Movassagh, Ramis. “Eigenvalue Attraction.” J Stat Phys 162, no. 3 (December 16, 2015): 615–643. https://orcid.org/0000-0002-4078-6752 en http://dx.doi.org/10.1007/s10955-015-1424-5 Journal of Statistical Physics Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. Springer Science+Business Media New York application/pdf Springer US Springer US |
| spellingShingle | Movassagh, Ramis Eigenvalue Attraction |
| title | Eigenvalue Attraction |
| title_full | Eigenvalue Attraction |
| title_fullStr | Eigenvalue Attraction |
| title_full_unstemmed | Eigenvalue Attraction |
| title_short | Eigenvalue Attraction |
| title_sort | eigenvalue attraction |
| url | http://hdl.handle.net/1721.1/106582 https://orcid.org/0000-0002-4078-6752 |
| work_keys_str_mv | AT movassaghramis eigenvalueattraction |