Conference abstracts

Session B5 - Random Matrices

July 14, 17:30 ~ 17:55

Eigenvalue Attraction

Ramis Movassagh

IBM TJ Watson Research Center, USA   -   q.eigenman@gmail.com

We prove that the complex conjugate (c.c.) eigenvalues of a smoothly varying real matrix attract. 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. We apply the results to various special cases including the Hatano-Nelson model and provide various 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.

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