The Exact Distribution of the Condition Number of Complex Random Matrices

Abstract EN

Let Gm×n (m≥n) be a complex random matrix and W=Gm×nHGm×n which is the complex Wishart matrix.

Let λ1>λ2>…>λn>0 and σ1>σ2>…>σn>0 denote the eigenvalues of the W and singular values of Gm×n, respectively.

The 2-norm condition number of Gm×n is κ2Gm×n=λ1/λn=σ1/σn.

In this paper, the exact distribution of the condition number of the complex Wishart matrices is derived.

The distribution is expressed in terms of complex zonal polynomials.