On the Law of Addition of Random Matrices

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 ^* 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.