Journal of Statistical Physics

, Volume 162, Issue 3, pp 615–643 | Cite as

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 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.


Complex Conjugate Real Axis Real Line Random Matrix Real Eigenvalue 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



I thank Leo P. Kadanoff, Steven G. Johnson, Tony Iarrobino and Gil Strang for discussions and the James Franck Institute at University of Chicago and the Perimeter Institute Canada, for having hosted me over the summer of 2013. I acknowledge the National Science Foundation’s support through Grant DMS. 1312831.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of MathematicsIBM T.J. Watson Research CenterYorktown HeightsUSA

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