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 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 kth largest eigenvalue finds applications in the performance
analysis of MIMO singular value decomposition systems.