Eigenvalue Attraction

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.

On Statistics of Bi-Orthogonal Eigenvectors in Real and Complex Ginibre Ensembles: Combining Partial Schur Decomposition with Supersymmetry

“…One motivation comes from the abovementioned relevance of eigenvector correlations for describing the motion of complex eigenvalues under perturbations of the ensemble, see e.g. [39], and associated Dysonian dynamics, see e.g. [5] and Appendix A of [3].…”

Eigenpairs of Toeplitz and Disordered Toeplitz Matrices with a Fisher–Hartwig Symbol

“…A general real matrix, under random real perturbations exhibits an attraction between any complex conjugate eigenvalues as proved elsewhere [39] . In [39] we did not rely from the onset on a Toeplitz structure nor perturbation theory and worked directly with spectral dynamics theory.…”

“…where quantities with the zero subscript denote the eigenvalues of the unperturbed problem (i.e., no disorder), that were analytically derived in Section II. Then, standard perturbation theory of non-Hermitian matrices (see for example Section 52 in [1,39]) to second order gives…”

Probing non-orthogonality of eigenvectors in non-Hermitian matrix models: diagrammatic approach

“…Second, the non-orthogonality of eigenfunctions was related to the statistics of resonance width shifts in open quantum systems [18], which was soon confirmed experimentally [19]. Third, the essential role of eigenvectors in stochastic motion of eigenvalues was revealed [20,21,22]. Last but not least, the topic has triggered the attention of the mathematical community [23,24].…”