# Asymptotic distribution of eigenvalues of random matrices and characterization of the Gaussian distribution by rotational invariance

1970

## Publisher

Virginia Polytechnic Institute and State University

## Abstract

The study falls in the area of random equations; in particular properties of random matrices have been studied. The dissertation makes precise some statistical theories of spectra developed in recent years by a number of physicists. Two basic results have been achieved.

The first result is a characterization of the distribution of a symmetric random matrix. Assuming independence of the diagonal and super-diagonal random variables of the symmetric random matrix the following theorem is proved: the distribution of the matrix is invariant under orthogonal similarity transforms if and only if the diagonal random variables are normally distributed with mean μ, and variance 2a², and the off-diagonal elements are normally distributed with mean O and variance a², :for some constants μ, and a² > O. The proof is achieved by solving a functional equation in characteristic functions. This seems to have been first proved in this context by Porter and Rosenzweig (Ann. Acad. Sci. Fennicae. AVI, No. 44, 1960) by a different method and under more restrictive conditions than those given here.

The second result deals with the asymptotic distribution of eigenvalues of a synnnetric random matrix as the dimension approaches infinity. Let An be an appropriately normalized n ⨉ n symmetric random matrix and let Wn(x) denote the empirical distribution function of the eigenvalues of An</sub. Under suitable conditions on the random variables of the matrix it is proved that Wn(x)⟶W(x) as n∞, where W is the absolutely continuous distribution function with a semi-circle density,

W(x) = {

⎧ 2/π (1-x²)1/2, |x| ≤ 1 ⎨ ⎩ 0 , |x| > 1.

The proof is achieved by an intricate combinatorial analysis in conjunction with the method of moments. This result extends a conjecture made by E. P. Wigner ("On the Distribution of the Roots of Certain Symmmetric Matrices," Ann. Math. 67, 1958, 325).