Hindawi Publishing Corporation The Scientific World Journal Volume 2013, Article ID 729839, 4 pages http://dx.doi.org/10.1155/2013/729839
Research Article The Exact Distribution of the Condition Number of Complex Random Matrices Lin Shi,1 Taibin Gan,2 Hong Zhu,1 and Xianming Gu2 1 2
School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China
Correspondence should be addressed to Lin Shi;
[email protected] Received 19 August 2013; Accepted 24 September 2013 Academic Editors: E. A. Abdel-Salam and E. Francomano Copyright Β© 2013 Lin Shi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. π» πΊπΓπ which is the complex Wishart matrix. Let π 1 > π 2 > . . . > π π > 0 Let πΊπΓπ (π β₯ π) be a complex random matrix and π = πΊπΓπ and π1 > π2 > . . . > ππ > 0 denote the eigenvalues of the W and singular values of πΊπΓπ , respectively. The 2-norm condition number of πΊπΓπ is π
2 (πΊπΓπ ) = βπ 1 /π π = π1 /ππ . 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.
1. 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 [1β13]. The capacity and performance of practical MIMO transmission schemes are often dictated by the statistical eigenproperties of the instantaneous channel π» πΊπΓπ , where πΊπΓπ is a complex correlation matrix π = πΊπΓπ Gaussian matrix and π 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 πΊπΓπ (π β₯ π) be a complex random matrix whose elements are independent and identically distributed (i.i.d.)
standard normal random variables. As we know, the π Γ π» πΊπΓπ is a complex π complex random matrix π = πΊπΓπ Wishart matrix. Its distribution is denoted by πβΌπ(π, Ξ£), Ξ£ = π2 πΌ. In addition, π is a positive definite Hermitian matrix with real eigenvalues; let π 1 > π 2 > β
β
β
> π π > 0 and π1 > π2 > β
β
β
> ππ > 0 denote the eigenvalues of the π and singular values of πΊπΓπ , respectively. The 2-norm condition number of πΊπΓπ is π
2 (πΊπΓπ ) = βπ 1 /π π = π1 /ππ ; thus, π
2 (π) = π
2 (πΊπΓπ )2 . The exact distributions of the condition number of a 2 Γ π 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 Ξ£ = π2 πΌ. 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 π
2 (ππΓπ ) for complex case, is proved.
2
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2. Some Preliminary Results
where
In this section, we give some results on joint density of the eigenvalues of a complex Wishart matrix π and πβΌπ(π, Ξ£), Ξ£ = π2 πΌ. The determinant, trace, and norm of a square matrix π΅ are denoted by |π΅|, tr(π΅), and βπ΅β, respectively. For any nonnegative integer π, a portion of π is a multiple (π1 , π2 , . . . , ππ ), where π1 β₯ π2 β₯ β
β
β
β₯ ππ β₯ 0 such that βππ=1 ππ = π, Dπ is the set of all portions of π, and the symbol βπ ππ
means the summation over Dπ ; that is, βπ ππ
= βπ
βDπ ππ
. Let π
be any portion of π, and let π1 > π2 > β
β
β
> ππ be the eigenvalues of an π Γ π matrix π΅ as follows: πΌ[π
] = π!
βππ 0 be the eigenvalues of π and Ξ = diag(π 1 , π 2 , . . . , π π ). If πβΌπ(π, Ξ£), Ξ£ = π2 πΌ with π β₯ π, then the joint density of its eigenvalues is defined [2] as follows:
π βππβπ β(π π π π=1 π π2 > β
β
β
> ππ > 0}, then [2]
π
1 1 [ π] = β( (π β π + 1)) , 2 π
π=1 2 ππ
πβπ
π where π‘ = π + π and π],π
is a constant coefficient.
where
π
(8)
πβDπ‘
π where π‘ = π + π and π],π
is a constant coefficient.
The following is the basic properties of the zonal polynomials [8]:
π (Ξ) =
π‘π₯β1 πβπ‘ ππ‘.
Lemma 2. For an π Γ π matrix π΅, the product of two zonal polynomials can be expressed in terms of a weighted combination of another zonal polynomial [7]; that is, for all ] β Dπ and βπ
β Dπ , one has
(1)
πΆπ
(π΅) = πΌ[π
] β
πΌ[π
] (π΅)
ππ(πβ1) (π2 )
(7)
π πΆ] (π΅) πΆπ
(π΅) = β π],π
πΆπ (π΅) ,
The zonal polynomials (also called Schur polynomials) of π΅ are defined by
(π₯)ππ = π₯ (π₯ + 1) β
β
β
(π₯ + ππ β 1) ,
+β
0
,
σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨[(πππ +πβπ )]σ΅¨σ΅¨σ΅¨ σ΅¨ σ΅¨σ΅¨ π πΌπ
(π΅) = σ΅¨ πβπ σ΅¨ σ΅¨ . σ΅¨σ΅¨[(π )]σ΅¨σ΅¨ σ΅¨σ΅¨ π σ΅¨σ΅¨
π
π
Ξπ (π) = ππ(πβ1)/2 βΞ (π + π + 1) ,
1 π β π π ) exp (β 2 βπ π ) , π π=1 2
(6)
The exact distribution of the 2-norm condition number of the Wishart matrix π is derived by the following. Theorem 5. Let πΊπΓπ be a complex random matrix and π = π» πΊπΓπ . π 1 and π π are the maximum and minimum πΊπΓπ
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3
eigenvalues of π, π
2 (π) = π 1 /π π . Then, the exact distribution of π
2 (π) is given by βππ
ππ(πβ1) (π2 )
π (π
2 (π)) =
Ξπ (π) Ξπ (π)
exp (β
π π ) π2 1
π=0 π
βDπ
1>πΎπβ1 >β
β
β
>πΎ2 >0
β π1 1 β β β (π β π)] π2π π! π =0 ]βD πβD π ! π
π
β«
ππβπ+π
β
Γβ β
By making the transformation ππ = πΎπ /πΎπ , π = 2, . . . , π β 1, using Lemma 3, and integrating over the set 1 > ππβ1 > β
β
β
> π2 > 0, we have
(πβ1)(π+1)+π+π β1
Γ π
2 (π)β(πβ1)(π+1)βπβπ +1 Γ (π2 β 1 + π + π )
(π )
exp (β
Ξπ (π) Ξπ (π)
1>ππβ1 >β
β
β
>π2 >0
π
π π ) πππβπ |π΄|2 |πΌ β π΄|πβπ π2 1 1
2
Γ β(πΎπ β πΎπ ) exp [ πππβ1 >β
β
β
>π2 >0
π
(π )
Ξπ (π) Ξπ (π) π
2
Γ β(πΎπ β πΎπ ) β β π β
β
β
> πΎ2 > 0, we obtain the joint distribution of π 1 and πΎ2 , . . . , πΎπ , as follows: π
Γβ«
Γ β(1 β ππ ) β (ππ β ππ ) βπππ ,
where π‘ = π + π.
2 βππ
π=2
π