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We consider a class of matrices of the form C n = (1/N )(R n +σX n )(R n +σX n ) * , where X n is an n × N matrix consisting of independent standardized complex entries, R j is an n×N nonrandom matrix, and σ > 0. Among several applications, C n can be viewed as a sample correlation matrix, where information is contained in (1/N )R n R * n , but each column of R n is contaminated by noise. As n → ∞, if n/N → c > 0, and the empirical distribution of the eigenvalues of (1/N )R n R * n converge to a proper probability distribution, then the empirical distribution of the eigenvalues of C n converges a.s. to a nonrandom limit. In this paper we show that, under certain conditions on R n , for any closed interval in R + outside the support of the limiting distribution, then, almost surely, no eigenvalues of C n will appear in this interval for all n large.

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.

A stronger result on the limiting distribution of the eigenvalues of random Hermitian matrices of the form A+XT X * , originally studied in Marčenko and Pastur [4], is presented. Here, X (N ×n), T (n×n), and A (N ×N) are independent, with X containing i.i.d. entries having finite second moments, T is diagonal with real (diagonal) entries, A is Hermitian, and n/N → c > 0 as N → ∞. Under addtional assumptions on the eigenvalues of A and T , almost sure convergence of the empirical distribution function of the eigenvalues of A + XT X * is proven with the aid of Stieltjes transforms, taking a more direct approach than previous methods.

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