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Eigenvalue Attraction

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Abstract

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.

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Notes

  1. We remark that the theory of pseudo-spectra [1] quantifies how far an eigenvalue can wander without quantifying the direction of the motion.

  2. Strictly speaking Dirac’s derivation of Eq. 10 in Sect. 43 of this reference, does not hold in general (e.g., non-Hermitian) as the left eigenvectors are not ’bras’ in his notation. The latter is a Hermitian conjugate of a standard (right) eigenvector. In his book, Dirac cites [Born, Heisenberg and Jordan, z.f. Physik 35, 565 (1925)] for these formulas.

  3. We denote the eigenvalue by \(\lambda _{k}\) instead of \(\lambda _{i}\) above because \(i\equiv \sqrt{-1}\) appears more in the following discussion.

  4. It is possible to construct \(C^{\infty }\) versions of such window functions such as the Planck-taper window function [27, pp. 127–134]; however, W has a simple form with the basic differentiability properties needed here.

  5. This question was posed to us by Freeman Dyson.

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Acknowledgments

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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  1. Department of Mathematics, IBM T.J. Watson Research Center, Yorktown Heights, NY, 10598, USA

    Ramis Movassagh

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Correspondence to Ramis Movassagh.

Appendix: Matlab Code

Appendix: Matlab Code

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Movassagh, R. Eigenvalue Attraction. J Stat Phys 162, 615–643 (2016). https://doi.org/10.1007/s10955-015-1424-5

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