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.Comment: v1:15 pages, 12 figures, 1 Matlab code. v2: very minor changes, fixed
a reference. v3: 25 pages, 17 figures and one Matlab code. The results have
been extended and generalized in various ways v4: 26 pages, 10 figures and a
Matlab Code. Journal Reference Added.
http://link.springer.com/article/10.1007%2Fs10955-015-1424-
