Complex singular wishart matrices and applications

In this paper, complex singular Wishart matrices and their applications to information theory are investigated. In particular, a volume element on the space of positive semidefinite m × m Hermitian matrices of rank n < m is introduced and some transformation properties are established. The Jacobian for the change of variables in the singular value decomposition of general m × n complex matrices is derived. Then, the density functions are formulated for all rank n complex singular Wishart distributions. From this, the joint eigenvalue density of low rank complex Wishart matrices are derived. The derived densities are used to evaluate the most important information-theoretic measure, the so-called ergodic channel capacity of multiple-input multiple-output (MIMO) spatially correlated Rayleigh distributed wireless communication channels.