Spectral Analysis of Random Matrices with a Rank One Pattern of Variances

Victor M. Preciado, Mohammad Amin Rahimian

Published 2014-09-18Version 1

We prove that the empirical spectral distributions for a class of random matrices, characterized by independent zero-mean entries and rank one pattern of variances, converge with probability one to a deterministic distribution, which is uniquely characterized by its sequence of moments, and we provide explicit expressions for these limiting moments. We next proffer efficient optimization programs that allow us to upper and lower bound the expected spectral norm of this class of random matrices for any finite $n$ and the almost sure limit of the spectral norm as $n \to \infty$. Asymptotic properties of random matrices with nonidentical entries is a challenging problem in the theory of random matrices, and bounds on their spectral moments have significance in a miscellany of applications.