Eigenvalues and Condition Numbers of Complex Random Matrices

Abstract

In this paper, the distributions of the largest and smallest eigenvalues of complex Wishart matrices and the condition number of complex Gaussian random matrices are derived. These distributions are represented by complex hypergeometric functions of matrix arguments, which can be expressed in terms of complex zonal polynomials. Several results are derived on complex hypergeometric functions and complex zonal polynomials and are used to evaluate these distributions. Finally, applications of these distributions in numerical analysis and statistical hypothesis testing are mentioned.