We investigate the dynamic behavior of the stationary random process defined by a central complex Wishart matrix @math as it varies along a certain dimension @math . We characterize the second-order joint cumulative distribution function (cdf) of the largest eigenvalue, and the second-order joint cdf of the smallest eigenvalue of this matrix. We show that both cdfs can be expressed in exact closed-form in terms of a finite number of well-known special functions in the context of communication theory. As a direct application, we investigate the dynamic behavior of the parallel channels associated with multiple-input multiple-output (MIMO) systems in the presence of Rayleigh fading. Studying the complex random matrix that defines the MIMO channel, we characterize the second-order joint cdf of the signal-to-noise ratio (SNR) for the best and worst channels. We use these results to study the rate of change of MIMO parallel channels, using different performance metrics. For a given value of the MIMO channel correlation coefficient, we observe how the SNR associated with the best parallel channel changes slower than the SNR of the worst channel. This different dynamic behavior is much more appreciable when the number of transmit ( @math ) and receive ( @math ) antennas is similar. However, as @math is increased while keeping @math fixed, we see how the best and worst channels tend to have a similar rate of change.
Since the seminal work by Wishart @cite_5 , random matrix theory has found application in very diverse fields like physics @cite_40 , neuroscience @cite_27 and many others @cite_33 . For instance, random matrix processes are useful in econometrics to study the stock volatility in portfolio management @cite_7 @cite_41 ; in immunology, random matrix theory has been used to design immunogens targeted for rapidly mutating viruses @cite_26 .
@cite_26 @cite_33 @cite_7 @cite_41 @cite_40 @cite_27 @cite_5
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Random matrix theory has found many applications in physics, statistics and engineering since its inception. Although early developments were motivated by practical experimental problems, random matrices are now used in fields as diverse as Riemann hypothesis, stochastic differential equations, condensed matter physics, statistical physics, chaotic systems, numerical linear algebra, neural networks, multivariate statistics, information theory, signal processing and small-world networks. This article provides a tutorial on random matrices which provides an overview of the theory and brings together in one source the most significant results recently obtained. Furthermore, the application of random matrix theory to the fundamental limits of wireless communication channels is described in depth. A fundamental result of free probability theory due to Voiculescu and subsequently refined by many authors states that conjugation by independent Haar-distributed random unitary matrices delivers asymptotic freeness. In this paper we exhibit many other systems of random unitary matrices that, when used for conjugation, lead to freeness. We do so by first proving a general result asserting “asymptotic liberation” under quite mild conditions, and then we explain how to specialize these general results in a striking way by exploiting Hadamard matrices. In particular, we recover and generalize results of the second-named author and of Tulino, Caire, Shamai and Verdu. Abstract We review the development of random-matrix theory (RMT) during the last fifteen years. We emphasize both the theoretical aspects, and the application of the theory to a number of fields. These comprise chaotic and disordered systems, the localization problem, many-body quantum systems, the Calogero-Sutherland model, chiral symmetry breaking in QCD, and quantum gravity in two dimensions. The review is preceded by a brief historical survey of the developments of RMT and of localization theory since their inception. We emphasize the concepts common to the above-mentioned fields as well as the great diversity of RMT. In view of the universality of RMT, we suggest that the current development signals the emergence of a new “statistical mechanics”: Stochasticity and general symmetry requirements lead to universal laws not based on dynamical principles. Random matrices now play a role in many areas of theoretical, applied,and computational mathematics. Therefore, it is desirable to have toolsfor studying random matrices that are flexible, easy to use, and powerful.Over the last fifteen years, researchers have developed a remarkablefamily of results, called matrix concentration inequalities, that achieveall of these goals.This monograph offers an invitation to the field of matrix concentrationinequalities. It begins with some history of random matrix theory;it describes a flexible model for random matrices that is suitablefor many problems; and it discusses the most important matrix concentrationresults. To demonstrate the value of these techniques, thepresentation includes examples drawn from statistics, machine learning,optimization, combinatorics, algorithms, scientific computing, andbeyond.