Isolated Eigenvalue of a Random Matrix

Abstract

A derivation is given of a property, noticed by Porter, of random real symmetric matrices. In an idealized physical problem, the random matrices were generated in a computer. If the elements of the matrices had a nonzero mean, then the matrices were found to have a statistically isolated eigenvalue. It is shown here that such an isolated eigenvalue exists for a large matrix if the mean value of the elements is a substantial fraction of the root mean square of the elements. The associated eigenvector has all its components close to equality. Counter examples are given, of matrices which are "statistical freaks," to show that the properties are statistical and not universal.