The Exact Distribution of the Condition Number of Complex Random Matrices

Let G m×n (m ≥ n) be a complex random matrix and W = G m×n H 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 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.


Introduction
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 [1][2][3][4][5][6][7][8][9][10][11][12][13]. The capacity and performance of practical MIMO transmission schemes are often dictated by the statistical eigenproperties of the instantaneous channel correlation matrix = × × , where × is a complex Gaussian matrix and is known to follow a complex Wishart distribution.
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 [7]). In nuclear physics, random eigenvalues are used to model nuclear energy levels and level spacing [6]. Moreover, the zeros of the Riemann zeta function are modeled using random eigenvalues [6]. Condition numbers arise in theory and applications of random matrices, such as multivariate statistics and quantum physics.
Let × ( ≥ ) be a complex random matrix whose elements are independent and identically distributed (i.i.d.) standard normal random variables. As we know, the × complex random matrix = × × is a complex Wishart matrix. Its distribution is denoted by ∼ ( , Σ), Σ = 2 . In addition, is a positive definite Hermitian matrix with real eigenvalues; let 1 > 2 > ⋅ ⋅ ⋅ > > 0 and 1 > 2 > ⋅ ⋅ ⋅ > > 0 denote the eigenvalues of the and singular values of × , respectively. The 2-norm condition number of × is 2 ( × ) = √ 1 / = 1 / ; thus, The exact distributions of the condition number of a 2 × matrix whose elements are independent and identically distributed standard normal real or complex random variables are given in [5] by Edelman. Edelman also obtained the limiting distributions and the limiting expected logarithms of the condition numbers of random rectangular matrices whose elements are independent and identically distributed standard normal random variables. The exact distributions of the condition number of Gaussian matrices are studied in [1] for real random matrix. Here, we derive the exact distribution of the condition number of a complex random matrix for special case Σ = 2 .
This paper is arranged as follows. Section 2 gives some preliminary results to the complex random matrices. In Section 3, the main result of this work, the density function of 2 ( × ) for complex case, is proved.

Some Preliminary Results
In this section, we give some results on joint density of the eigenvalues of a complex Wishart matrix and ∼ ( , Σ), Σ = 2 . The determinant, trace, and norm of a square matrix are denoted by | |, tr( ), and ‖ ‖, respectively. For any nonnegative integer , a portion of is a multiple = , D is the set of all portions of , and the symbol ∑ means the summation over D ; that is, ∑ = ∑ ∈D .
Let be any portion of , and let 1 > 2 > ⋅ ⋅ ⋅ > be the eigenvalues of an × matrix as follows: The zonal polynomials (also called Schur polynomials) of are defined by The following is the basic properties of the zonal polynomials [8]: The zonal polynomial of the identity matrix is defined by where where is positive definite Hermitian matrix with real eigenvalues, let × be a complex random matrix whose elements are independent and identically distributed (i.i.d.) standard normal random variables. Let 1 > 2 > ⋅ ⋅ ⋅ > > 0 be the eigenvalues of and Λ = diag( 1 , 2 , . . . , ). If ∼ ( , Σ), Σ = 2 with ≥ , then the joint density of its eigenvalues is defined [2] as follows: where Γ ( ) = ( −1)/2 ∏ =1 Γ ( + + 1) , Lemma 2. For an × matrix , the product of two zonal polynomials can be expressed in terms of a weighted combination of another zonal polynomial [7]; that is, for all ] ∈ D and ∀ ∈ D , one has where = + and ], is a constant coefficient.

Lemma 3. For an × matrix and for all ∈ D , one has [1]
where = + and ], is a constant coefficient. where

Main Result
The exact distribution of the 2-norm condition number of the Wishart matrix is derived by the following.
Theorem 5. Let × be a complex random matrix and = × × . 1 and are the maximum and minimum eigenvalues of , 2 ( ) = 1 / . Then, the exact distribution of 2 ( ) is given by where = + .