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Let  G    m × n   ( m  ≥  n ) be a complex random matrix and  W  =  G    m × nH    G    m × n   which is the complex Wishart matrix. Let  λ   1  >  λ   2  > …> λ    n   > 0 and  σ   1  >  σ   2  > …> σ    n   > 0 denote the eigenvalues of the  W  and singular values of  G    m × n  , respectively. The 2-norm condition number of  G    m × n   isκ    2G    m  ×  n=λ    1    /    λ    n=    σ    1    /    σ    n. 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.

The Exact Distribution of the Condition Number of Complex Random Matrices