# The Exact Distribution of the Condition Number of Complex Random Matrices

Authors: Lin Shi; Taibin Gan; Hong Zhu; Xianming Gu
Journal: The Scientific World Journal
 (2013)

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## Abstract

LetGm×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.


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## Body

## 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 [1–13]. The capacity and performance of practical MIMO transmission schemes are often dictated by the statistical eigenproperties of the instantaneous channel correlation matrix W=Gm×nHGm×n, where Gm×n is a complex Gaussian matrix and W 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.LetGm×n(m≥n) be a complex random matrix whose elements are independent and identically distributed (i.i.d.) standard normal random variables. As we know, the n×n complex random matrix W=Gm×nHGm×n is a complex Wishart matrix. Its distribution is denoted by W~W(m,Σ), Σ=σ2I. In addition, W is a positive definite Hermitian matrix with real eigenvalues; 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 κ2(Gm×n)=λ1/λn=σ1/σn; thus, κ2(W)=κ2(Gm×n)2.The exact distributions of the condition number of a2×n 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=σ2I.This paper is arranged as follows. Section2 gives some preliminary results to the complex random matrices. In Section 3, the main result of this work, the density function of κ2(Wn×n) for complex case, is proved.


## 2. Some Preliminary Results

In this section, we give some results on joint density of the eigenvalues of a complex Wishart matrixW and W~W(m,Σ), Σ=σ2I. The determinant, trace, and norm of a square matrix B are denoted by |B|, tr(B), and ∥B∥, respectively.For any nonnegative integerp, a portion of p is a multiple (a1,a2,…,an), where a1≥a2≥⋯≥an≥0 such that ∑i=1nai=p, 𝒟p is the set of all portions of p, and the symbol ∑paκ means the summation over 𝒟p; that is, ∑paκ=∑κ∈𝒟paκ.Letκ be any portion of p, and let μ1>μ2>⋯>μn be the eigenvalues of an n×n matrix B as follows:
(1)α[κ]=p!∏i<jn(ai-aj-i+j)∏i<jn‍(ai+n-i)!,ακ(B)=|[(μiaj+n-j)]||[(μin-j)]|.
The zonal polynomials (also called Schur polynomials) of B are defined by
(2)Cκ(B)=α[κ]·α[κ](B)
The following is the basic properties of the zonal polynomials [8]:
(3)[tr(B)]p=∑pCκ(B)=∑κ∈𝒟pCκ(B).The zonal polynomial of the identity matrix is defined by(4)Cκ(In)=22pp![12n]κ∏i<jr(2ai-2aj-i+j)∏1r‍(2ai+r-i)!,
where
(5)[12n]κ=∏i=1n(12(n-i+1))ai,(x)ai=x(x+1)⋯(x+ai-1),(x)0=1.Definition 1.
ForW=Gm×nHGm×n, where W is positive definite Hermitian matrix with real eigenvalues, let Gm×n be a complex random matrix whose elements are independent and identically distributed (i.i.d.) standard normal random variables. Let λ1>λ2>⋯>λn>0 be the eigenvalues of W and Λ=diag(λ1,λ2,…,λn). If W~W(m,Σ), Σ=σ2I with m≥n, then the joint density of its eigenvalues is defined [2] as follows:
(6)f(Λ)=πn(n-1)(σ2)-mnΓn(n)Γn(m)×∏i=1nλim-n∏i<ln(λi-λl)2exp(-1σ2∑i=1nλi),
where
(7)Γn(a)=πn(n-1)/2∏i=1nΓ(a+k+1),Γ(x)=∫0+∞tx-1e-tdt.Lemma 2.
For ann×n matrix B, the product of two zonal polynomials can be expressed in terms of a weighted combination of another zonal polynomial [7]; that is, for all ν∈𝒟s and ∀κ∈𝒟p, one has
(8)Cν(B)Cκ(B)=∑τ∈𝒟tgν,κτCτ(B),
where t=s+p and gν,κτ is a constant coefficient.Lemma 3.
For anm×n matrix B and for all κ∈𝒟p, one has [1]
(9)|I-B|m-nCκ(B)=∑s=0∞∑ν∈𝒟s∑τ∈𝒟t(n-m)νgν,κτCτ(B)s!,
where t=s+p and gν,κτ is a constant coefficient.Lemma 4.
LetZ=diag(ξ2,…,ξn), Z1=diag(1,ξ2,…,ξn) and D={1>ξ2>⋯>ξn>0}, then [2]
(10)∫D|Z|b-n∏i=2n(1-ξi)2∏i<ln(ξi-ξl)2Cκ(Z1)∏i=2ndξi=(nb+p)Γn(n)πn(n-1)Γn(b,κ)Γn(n)Γn(b+n,κ)Cκ(I),
where
(11)Γn(a)=πn(n-1)/2∏i=1nΓ(a+k+1),Γn(b,κ)=πn(n-1)/2∏i=1nΓ(a+ki-i+1).


## 3. Main Result

The exact distribution of the 2-norm condition number of the Wishart matrixW is derived by the following.Theorem 5.
LetGm×n be a complex random matrix and W=Gm×nHGm×n. λ1 and λn are the maximum and minimum eigenvalues of W, κ2(W)=λ1/λn. Then, the exact distribution of κ2(W) is given by
(12)f(κ2(W))=πn(n-1)(σ2)-mnΓn(n)Γn(m)exp(-nσ2λ1)×∑p=0∞∑κ∈𝒟pλ1mn-n+pσ2pp!∑s=0∞∑ν∈𝒟s∑τ∈𝒟t1s!(n-m)ν×gν,κτCτ(κ2(W)-1)(n-1)(n+1)+p+s-1×κ2(W)-(n-1)(n+1)-p-s+1×(n2-1+p+s)×Γn-1(n-1)π(n-1)(n-2)×Γn-1(n+1,τ)Γn-1(n-1)Γn-1(2n,τ)Cτ(I),
where t=s+p.Proof (consider (6)).
By making the transformationλ1=λ1, γi=1-λi/λ1, i=2,…,n, where 1>γn>⋯>γ2>0, we obtain the joint  distribution of λ1 and γ2,…,γn, as follows:
(13)πn(n-1)(σ2)-mnΓn(n)Γn(m)exp(-nσ2λ1)λ1mn-n|A|2|I-A|m-n×∏k<ln(γk-γk)2exp[λ1σ2tr(A)].
By using Taylor’s formula,(14)πn(n-1)(σ2)-mnΓn(n)Γn(m)exp(-nσ2λ1)λ1mn-n|A|2|I-A|m-n×∏k<ln(γk-γk)2∑p=0∞1p![λ1σ2tr(A)]p.
By using property (3), we have
(15)πn(n-1)(σ2)-mnΓn(n)Γn(m)exp(-nσ2λ1)|A|2|I-A|m-n×∏i<ln(γi-γl)2∑p=0∞∑κ∈𝒟pλ1mn-n+pCκ(A)σ2pp!,
where A=diag(γ2,…,γn).
By making the transformationξi=γi/γn, i=2,…,n-1, using Lemma 3, and integrating over the set 1>ξn-1>⋯>ξ2>0, we have
(16)∫1>γn-1>⋯>γ2>0|A|2|I-A|m-nCκ(A)∏i<ln(γi-γl)2∏i=2n-1dγi=[∑s=0∞∑ν∈𝒟s∑τ∈𝒟t1s!(n-m)νgν,κτγn(n-1)(n-2)+p+s-1]×∫1>ξn-1>⋯>ξ2>0|Z|2Cτ(Z1)×∏i=2n-1(1-ξi)2∏i,j=2n-1(ξi-ξj)2∏i=2n-1dξi,
where Z=diag(ξ2,…,ξn-1) and Z1=diag(1,Z).
By using Lemma4, let n be replaced by n-1, let κ be replaced by τ, and let b=n+1.
Then,(17)∫1>ξn-1>⋯>ξ2>0|Z|2Cτ(Z1)×∏i=2n-1(1-ξi)2∏i,j=2n-1(ξi-ξj)2∏i=2n-1dξi=(n2-1+p+s)Γn-1(n-1)πn(n-1)×Γn-1(n+1,τ)Γn-1(n-1)Γn-1(2n,τ)Cτ(I).
It follows that the distribution of(λ1,γn) is given by
(18)f(λ1,γn)=πn(n-1)(σ2)-mnΓn(n)Γn(m)exp(-nσ2λ1)×∑p=0∞∑κ∈𝒟pλ1mn-n+pσ2pp!∑s=0∞∑ν∈𝒟s∑τ∈𝒟t1s!(n-m)ν×gν,κτγn(n-1)(n+1)+p+s-1(n2-1+p+s)×Γn-1(n-1)π(n-1)(n-2)Γn-1(n+1,τ)Γn-1(n-1)Γn-1(2n,τ)Cτ(I).
Note thatκ2(W)=λ1/λn=(1-γn)-1; then, the distribution of κ2(W)=(1-γn)-1 is given by
(19)f(κ2(W))=πn(n-1)(σ2)-mnΓn(n)Γn(m)exp(-nσ2λ1)×∑p=0∞∑κ∈𝒟pλ1mn-n+pσ2pp!∑s=0∞∑ν∈𝒟s∑τ∈𝒟t1s!(n-m)ν×gν,κτCτ(κ2(W)-1)(n-1)(n+1)+p+s-1×κ2(W)-(n-1)(n+1)-p-s+1×(n2-1+p+s)×Γn-1(n-1)π(n-1)(n-2)×Γn-1(n+1,τ)Γn-1(n-1)Γn-1(2n,τ)Cτ(I).


## 4. Conclusions

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.



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