Spectral Properties of Random Non-self-adjoint Matrices and Operators

TL;DR

This paper investigates the spectral instability of random non-self-adjoint matrices and operators, revealing intrinsic uncomputability of eigenvalues and complex spectral behavior in infinite-dimensional limits.

Contribution

It provides numerical evidence of spectral instability in random matrices and introduces a stochastic operator family illustrating spectral complexity in infinite dimensions.

Findings

Eigenvalues are likely uncomputable for large random matrices.

Eigenvectors can generate dense subspaces despite spectral ambiguity.

Spectral properties change abruptly in the infinite volume limit.

Abstract

We describe some numerical experiments which determine the degree of spectral instability of medium size randomly generated matrices which are far from self-adjoint. The conclusion is that the eigenvalues are likely to be intrinsically uncomputable for similar matrices of a larger size. We also describe a stochastic family of bounded operators in infinite dimensions for almost all of which the eigenvectors generate a dense linear subspace, but the eigenvalues do not determine the spectrum. Our results imply that the spectrum of the non-self-adjoint Anderson model changes suddenly as one passes to the infinite volume limit.