On the Marginal Distribution of the Eigenvalues of Wishart Matrices

Random matrices play a crucial role in the design and analysis of multiple-input multiple-output (MIMO) systems. In particular, performance of MIMO systems depends on the statistical properties of a subclass of random matrices known as Wishart when the propagation environment is characterized by Rayleigh or Rician fading. This paper focuses on the stochastic analysis of this class of matrices and proposes a general methodology to evaluate some multiple nested integrals of interest. With this methodology we obtain a closed-form expression for the joint probability density function of k consecutive ordered eigenvalues and, as a special case, the PDF of the l(th) ordered eigenvalue of Wishart matrices. The distribution of the largest eigenvalue can be used to analyze the performance of MIMO maximal ratio combining systems, The PDF of the smallest eigenvalue can be used for MIMO antenna selection techniques. Finally, the PDF the k(th) largest eigenvalue finds applications in the performance analysis of MIMO singular value decomposition systems.