^{1}

^{2}

^{1}

^{2}

^{1}

^{2}

Let

Over the past decade, multiple-input and multiple-output (MIMO) systems have been at the forefront of wireless communications research and development, due to their huge potential for delivering significant capacity compared with conventional systems [

In recent years, the statistical properties of Wishart matrices have been extensively studied and applied to a large number of MIMO applications. In statistics, the random eigenvalues are used in hypothesis testing, principal component analysis, canonical correlation analysis, multiple discriminant analysis, and so forth (see [

Let

The exact distributions of the condition number of a

This paper is arranged as follows. Section

In this section, we give some results on joint density of the eigenvalues of a complex Wishart matrix

For any nonnegative integer

Let

The zonal polynomial of the identity matrix is defined by

For

For an

For an

Let

The exact distribution of the 2-norm condition number of the Wishart matrix

Let

By making the transformation

By using Taylor’s formula,

By using property (

By making the transformation

By using Lemma

Then,

It follows that the distribution of

Note that

In this paper, the exact distribution of the condition number of complex Wishart matrices is derived. The distribution is expressed in terms of complex zonal polynomials. This distribution plays an important role in numerical analysis and statistical hypothesis testing.

The authors would like to thank the anonymous reviewers for their detailed comments and suggestions. This work was supported by the Fund of Sichuan Provincial Key Laboratory of Signal and Information Processing, Xihua University (SZJJ2009-002).