Random matrix theory
Abstract
Random matrix theory deals with the study of matrix-valued random variables. It is conventionally considered that random matrix theory dates back to the work of Wishart in 1928 [1] on the properties of matrices of the type XX † with X ε ℂ N×n a random matrix with independent Gaussian entries with zero mean and equal variance. Wishart and his followers were primarily interested in the joint distribution of the entries of such matrices and then on their eigenvalues distribution. It then dawned to mathematicians that, as the matrix dimensions N and n grow large with ratio converging to a positive value, its eigenvalue distribution converges weakly and almost surely to some deterministic distribution, which is somewhat similar to a law of large numbers for random matrices. This triggered a growing interest in particular among the signal processing community, as it is usually difficult to deal efficiently with large dimensional data because of the so-called curse of dimensionality. Other fields of research have been interested in large dimensional random matrices, among which the field of wireless communications, as the eigenvalue distribution of some random matrices is often a sufficient statistics for the performance evaluation of multidimensional wireless communication systems. © 2011 by Taylor and Francis Group, LLC.
Keywords
Eigenvalues and eigenfunctions
Random variables
Signal processing
Wireless telecommunication systems
Curse of dimensionality
Eigenvalue distributions
Eigenvalues distribution
Law of large numbers
Performance evaluations
Sufficient statistics
Wireless communication system
Wireless communications
Matrix algebra