Volume 12, Number 2, Summer 2004
JACOBIANS AND HYPERGEOMETRIC
FUNCTIONS IN COMPLEX MULTIVARIATE
T. RATNARAJAH, R. VAILLANCOURT AND M. ALVO
Abstract. In this paper, the Jacobians for complex matrix
transformations are derived by means of the exterior product.
The transformations are Cholesky factorization, eigen-decomposition
and orthonormal-triangular factorization, which
frequently occur in complex multivariate analysis. As an example,
using these Jacobians we derive the volume of Stiefel manifold
and the complex noncentral Wishart density. Moreover,
the complex multivariate densities are often represented by
complex hypergeometric functions of matrix arguments, which
can be expressed in terms of complex zonal polynomials. We
give a method to compute these complex hypergeometric functions.
Finally, applications of these complex random matrix
theories to information theory and numerical analysis are mentioned.