On the Law of Addition of Random Matrices

TL;DR

This paper investigates the eigenvalue distribution of the sum of two large, rotated Hermitian matrices, establishing convergence to a deterministic measure and deriving a functional equation for its Stieltjes transform.

Contribution

It introduces a new functional equation relating the eigenvalue measures of individual matrices to their sum in the large size limit.

Findings

Eigenvalue measures converge in probability to a deterministic limit.

A functional equation for the Stieltjes transform of the limiting measure is derived.

The results extend understanding of eigenvalue distributions under random rotations.

Abstract

Normalized eigenvalue counting measure of the sum of two Hermitian (or real symmetric) matrices $A_{n}$ and $B_{n}$ rotated independently with respect to each other by the random unitary (or orthogonal) Haar distributed matrix $U_{n}$ (i.e. $A_{n}+U_{n}^{\ast}B_{n}U_{n}$) is studied in the limit of large matrix order $n$. Convergence in probability to a limiting nonrandom measure is established. A functional equation for the Stieltjes transform of the limiting measure in terms of limiting eigenvalue measures of $A_{n}$ and $B_{n}$ is obtained and studied. Keywords: random matrices, eigenvalue distribution