Statistica Sinica

Z. D. Bai

Abstract:In this paper, we give a brief review of the theory of spectral analysis of large dimensional random matrices. Most of the existing work in the literature has been stated for real matrices but the corresponding results for the complex case are also of interest, especially for researchers in Electrical and Electronic Engineering. Thus, we convert almost all results to the complex case, whenever possible. Only the latest results, including some new ones, are stated as theorems here. The main purpose of the paper is to show how important methodologies, or mathematical tools, have helped to develop the theory. Some unsolved problems are also stated.

Key words and phrases:Circular law, complex random matrix, largest and smallest eigenvalues of a random matrix, noncentral Hermitian matrix, spectral analysis of large dimensional random matrices, spectral radius.