On the Distribution of the Roots of Certain Symmetric Matrices

TLDR

The article studies the distribution of eigenvalues of very high‑dimensional real symmetric matrices. The study shows that a previously derived eigenvalue distribution law for a special matrix set extends to broader classes. Matrices are N×N real symmetric with elements a_{ij} satisfying a_{ij}=a_{ji}.

Abstract

The present article is concerned with the distribution of the latent roots (characteristic values) of certain sets of real symmetric matrices of very high dimensionality. Its purpose is to point out that the distribution law obtained before' for a very special set of matrices is valid for much more general sets. The dimension of the matrices will be denoted by N, the matrix elements by via. These are real. The condition of symmetry is