The Exact Distribution of the Condition Number of Complex Random Matrices

Abstract

Let 𝐺_𝑚×𝑛(𝑚≥𝑛) be a complex random matrix and 𝑊=𝐺^𝐻_𝑚×𝑛𝐺_𝑚×𝑛 which is the complex Wishart matrix. Let 𝜆₁>𝜆₂>…>𝜆_𝑛>0 and 𝜎₁>𝜎₂>…>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 𝜅₂(𝐺_𝑚×𝑛)=𝜆₁/𝜆_𝑛=𝜎₁/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.

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=𝐺^𝐻_𝑚×𝑛𝐺_𝑚×𝑛, where 𝐺_𝑚×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.

Let 𝐺_𝑚×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=𝐺^𝐻_𝑚×𝑛𝐺_𝑚×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>⋯>𝜆_𝑛>0 and 𝜎₁>𝜎₂>⋯>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 𝜅₂(𝐺_𝑚×𝑛)=𝜆₁/𝜆_𝑛=𝜎₁/𝜎_𝑛; thus, 𝜅₂(𝑊)=𝜅₂(𝐺_𝑚×𝑛)².

The exact distributions of the condition number of a 2×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. Section 2 gives some preliminary results to the complex random matrices. In Section 3, the main result of this work, the density function of 𝜅₂(𝑊_𝑛×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 matrix 𝑊 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 integer 𝑝, a portion of 𝑝 is a multiple (𝑎₁,𝑎₂,…,𝑎_𝑛), where 𝑎₁≥𝑎₂≥⋯≥𝑎_𝑛≥0 such that ∑^𝑛_𝑖=1𝑎_𝑖=𝑝, 𝒟_𝑝 is the set of all portions of 𝑝, and the symbol ∑_𝑝𝑎_𝜅 means the summation over 𝒟_𝑝; that is, ∑_𝑝𝑎_𝜅=∑_{𝜅∈𝒟𝑝}𝑎_𝜅.

Let 𝜅 be any portion of 𝑝, and let 𝜇₁>𝜇₂>⋯>𝜇_𝑛 be the eigenvalues of an 𝑛×𝑛 matrix 𝐵 as follows: 𝛼_[𝜅]∏=𝑝!^{𝑛𝑖<𝑗}𝑎_𝑖−𝑎_𝑗−𝑖+𝑗∏^𝑛_𝑖<𝑗𝑎_𝑖!,𝛼+𝑛−𝑖_𝜅(𝐵)=𝜇^𝑎𝑗_𝑖^+𝑛−𝑗𝜇_𝑖^𝑛−𝑗. (1) The zonal polynomials (also called Schur polynomials) of 𝐵 are defined by 𝐶_𝜅(𝐵)=𝛼_[𝜅]⋅𝛼_[𝜅](𝐵) (2) The following is the basic properties of the zonal polynomials [8]: [tr(𝐵)]^𝑝=_𝑝𝐶_𝜅(𝐵)=_{𝜅∈𝒟𝑝}𝐶_𝜅(𝐵). (3)

The zonal polynomial of the identity matrix is defined by 𝐶_𝜅𝐼_𝑛=2^2𝑝1𝑝!2𝑛_𝜅∏^𝑟_𝑖<𝑗2𝑎_𝑖−2𝑎_𝑗−𝑖+𝑗∏^𝑟₁2𝑎_𝑖!,+𝑟−𝑖 (4) where 12𝑛_𝜅=^𝑛_𝑖=112(𝑛−𝑖+1)_𝑎𝑖,(𝑥)_𝑎𝑖=𝑥(𝑥+1)⋯𝑥+𝑎_𝑖−1,(𝑥)₀=1. (5)

Definition 1. For 𝑊=𝐺^𝐻_𝑚×𝑛𝐺_𝑚×𝑛, where 𝑊 is positive definite Hermitian matrix with real eigenvalues, let 𝐺_𝑚×𝑛 be a complex random matrix whose elements are independent and identically distributed (i.i.d.) standard normal random variables. Let 𝜆₁>𝜆₂>⋯>𝜆_𝑛>0 be the eigenvalues of 𝑊 and Λ=diag(𝜆₁,𝜆₂,…,𝜆_𝑛). If 𝑊∼𝑊(𝑚,Σ), Σ=𝜎²𝐼 with 𝑚≥𝑛, then the joint density of its eigenvalues is defined [2] as follows: 𝑓(Λ)=𝜋^{𝑛(𝑛−1)}𝜎^2−𝑚𝑛Γ_𝑛(𝑛)Γ_𝑛(𝑚)×^𝑛_𝑖=1𝜆_𝑖^{𝑛𝑚−𝑛}_𝑖<𝑙𝜆_𝑖−𝜆_𝑙²−1exp𝜎^2𝑛_𝑖=1𝜆_𝑖, (6) where Γ_𝑛(𝑎)=𝜋^{𝑛𝑛(𝑛−1)/2}_𝑖=1Γ(),Γ(𝑥)=𝑎+𝑘+1_0+∞𝑡^𝑥−1𝑒^−𝑡𝑑𝑡. (7)

Lemma 2. For an 𝑛×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∈𝒟_𝑠 and ∀𝜅∈𝒟_𝑝, one has 𝐶_𝜈(𝐵)𝐶_𝜅(𝐵)=_{𝜏∈𝒟𝑡}𝑔^𝜏_𝜈,𝜅𝐶_𝜏(𝐵), (8) where 𝑡=𝑠+𝑝 and 𝑔^𝜏_𝜈,𝜅 is a constant coefficient.

Lemma 3. For an 𝑚×𝑛 matrix 𝐵 and for all 𝜅∈𝒟_𝑝, one has [1] |𝐼−𝐵|^𝑚−𝑛𝐶_𝜅(𝐵)=∑^∞_𝑠=0∑_{𝜈∈𝒟𝑠}∑_{𝜏∈𝒟𝑡}(𝑛−𝑚)_𝜈𝑔^𝜏_𝜈,𝜅𝐶_𝜏(𝐵),𝑠! (9) where 𝑡=𝑠+𝑝 and 𝑔^𝜏_𝜈,𝜅 is a constant coefficient.

Lemma 4. Let 𝑍=diag(𝜉₂,…,𝜉_𝑛), 𝑍₁=diag(1,𝜉₂,…,𝜉_𝑛) and 𝐷={1>𝜉₂>⋯>𝜉_𝑛>0}, then [2] _𝐷|𝑍|^{𝑛𝑏−𝑛}_𝑖=21−𝜉_𝑖^2𝑛_𝑖<𝑙𝜉_𝑖−𝜉_𝑙²𝐶_𝜅𝑍₁^𝑛_𝑖=2𝑑𝜉_𝑖=Γ𝑛𝑏+𝑝^𝑛(𝑛)𝜋^{𝑛(𝑛−1)}Γ_𝑛(𝑏,𝜅)Γ_𝑛(𝑛)Γ_𝑛()𝐶𝑏+𝑛,𝜅^𝜅(𝐼), (10) where Γ_𝑛(𝑎)=𝜋^{𝑛𝑛(𝑛−1)/2}_𝑖=1Γ(),Γ𝑎+𝑘+1_𝑛(𝑏,𝜅)=𝜋^{𝑛𝑛(𝑛−1)/2}_𝑖=1Γ𝑎+𝑘_𝑖−𝑖+1. (11)

3. Main Result

The exact distribution of the 2-norm condition number of the Wishart matrix 𝑊 is derived by the following.

Theorem 5. Let 𝐺_𝑚×𝑛 be a complex random matrix and 𝑊=𝐺^𝐻_𝑚×𝑛𝐺_𝑚×𝑛. 𝜆₁ and 𝜆_𝑛 are the maximum and minimum eigenvalues of 𝑊, 𝜅₂(𝑊)=𝜆₁/𝜆_𝑛. Then, the exact distribution of 𝜅₂(𝑊) is given by 𝑓𝜅₂(𝑊)=𝜋^{𝑛(𝑛−1)}𝜎^2−𝑚𝑛Γ_𝑛(𝑛)Γ_𝑛(𝑚)−𝑛exp𝜎²𝜆₁×^∞_{𝑝=0𝜅∈𝒟𝑝}𝜆₁^{𝑚𝑛−𝑛+𝑝}𝜎^2𝑝𝑝!^∞_{𝑠=0𝜈∈𝒟𝑠𝜏∈𝒟𝑡}1()𝑠!𝑛−𝑚_𝜈×𝑔^𝜏_𝜈,𝜅𝐶_𝜏𝜅₂(𝑊)−1^{(𝑛−1)(𝑛+1)+𝑝+𝑠−1}×𝜅₂(𝑊)^{−(𝑛−1)(𝑛+1)−𝑝−𝑠+1}×𝑛²×Γ−1+𝑝+𝑠_𝑛−1(𝑛−1)𝜋^{(𝑛−1)(𝑛−2)}×Γ_𝑛−1(𝑛+1,𝜏)Γ_𝑛−1(𝑛−1)Γ_𝑛−1()𝐶2𝑛,𝜏^𝜏(𝐼), (12) where 𝑡=𝑠+𝑝.

Proof (consider (6)). By making the transformation 𝜆₁=𝜆₁, 𝛾_𝑖=1−𝜆_𝑖/𝜆₁, 𝑖=2,…,𝑛, where 1>𝛾_𝑛>⋯>𝛾₂>0, we obtain the joint distribution of 𝜆₁ and 𝛾₂,…,𝛾_𝑛, as follows: 𝜋^{𝑛(𝑛−1)}𝜎^2−𝑚𝑛Γ_𝑛(𝑛)Γ_𝑛(𝑚)−𝑛exp𝜎²𝜆₁𝜆₁^{𝑚𝑛−𝑛}|𝐴|²|𝐼−𝐴|^𝑚−𝑛×^𝑛_𝑘<𝑙𝛾_𝑘−𝛾_𝑘²𝜆exp¹𝜎²tr(𝐴). (13)
By using Taylor’s formula, 𝜋^{𝑛(𝑛−1)}𝜎^2−𝑚𝑛Γ_𝑛(𝑛)Γ_𝑛(𝑚)−𝑛exp𝜎²𝜆₁𝜆₁^{𝑚𝑛−𝑛}|𝐴|²|𝐼−𝐴|^𝑚−𝑛×^𝑛_𝑘<𝑙𝛾_𝑘−𝛾_𝑘^2∞_𝑝=01𝜆𝑝!𝜎¹²tr(𝐴)^𝑝. (14)
By using property (3), we have 𝜋^{𝑛(𝑛−1)}𝜎^2−𝑚𝑛Γ_𝑛(𝑛)Γ_𝑛(𝑚)−𝑛exp𝜎²𝜆₁|𝐴|²|𝐼−𝐴|^𝑚−𝑛×^𝑛_𝑖<𝑙𝛾_𝑖−𝛾_𝑙^2∞_{𝑝=0𝜅∈𝒟𝑝}𝜆₁^{𝑚𝑛−𝑛+𝑝}𝐶_𝜅(𝐴)𝜎^2𝑝,𝑝! (15) where 𝐴=diag(𝛾₂,…,𝛾_𝑛).
By making the transformation 𝜉_𝑖=𝛾_𝑖/𝛾_𝑛, 𝑖=2,…,𝑛−1, using Lemma 3, and integrating over the set 1>𝜉_𝑛−1>⋯>𝜉₂>0, we have _{1>𝛾𝑛−1>⋯>𝛾2>0}|𝐴|²|𝐼−𝐴|^𝑚−𝑛𝐶_𝜅(𝐴)^𝑛_𝑖<𝑙𝛾_𝑖−𝛾_𝑙^2𝑛−1_𝑖=2𝑑𝛾_𝑖=^∞_{𝑠=0𝜈∈𝒟𝑠𝜏∈𝒟𝑡}1()𝑠!𝑛−𝑚_𝜈𝑔^𝜏_𝜈,𝜅𝛾_𝑛^{(𝑛−1)(𝑛−2)+𝑝+𝑠−1}×_{1>𝜉𝑛−1>⋯>𝜉2>0}|𝑍|²𝐶_𝜏𝑍₁×^𝑛−1_𝑖=21−𝜉_𝑖^2𝑛−1_𝑖,𝑗=2𝜉_𝑖−𝜉_𝑗^2𝑛−1_𝑖=2𝑑𝜉_𝑖, (16) where 𝑍=diag(𝜉₂,…,𝜉_𝑛−1) and 𝑍₁=diag(1,𝑍).
By using Lemma 4, let 𝑛 be replaced by 𝑛−1, let 𝜅 be replaced by 𝜏, and let 𝑏=𝑛+1.
Then, _{1>𝜉𝑛−1>⋯>𝜉2>0}|𝑍|²𝐶_𝜏𝑍₁×^𝑛−1_𝑖=21−𝜉_𝑖^2𝑛−1_𝑖,𝑗=2𝜉_𝑖−𝜉_𝑗^2𝑛−1_𝑖=2𝑑𝜉_𝑖=𝑛²Γ−1+𝑝+𝑠^𝑛−1(𝑛−1)𝜋^{𝑛(𝑛−1)}×Γ_𝑛−1(𝑛+1,𝜏)Γ_𝑛−1(𝑛−1)Γ_𝑛−1()𝐶2𝑛,𝜏^𝜏(𝐼). (17)
It follows that the distribution of (𝜆₁,𝛾_𝑛) is given by 𝑓𝜆₁,𝛾_𝑛=𝜋^{𝑛(𝑛−1)}𝜎^2−𝑚𝑛Γ_𝑛(𝑛)Γ_𝑛(𝑚)−𝑛exp𝜎²𝜆₁×^∞_{𝑝=0𝜅∈𝒟𝑝}𝜆₁^{𝑚𝑛−𝑛+𝑝}𝜎^2𝑝𝑝!^∞_{𝑠=0𝜈∈𝒟𝑠𝜏∈𝒟𝑡}1()𝑠!𝑛−𝑚_𝜈×𝑔^𝜏_𝜈,𝜅𝛾_𝑛^{(𝑛−1)(𝑛+1)+𝑝+𝑠−1}𝑛²×Γ−1+𝑝+𝑠_𝑛−1(𝑛−1)𝜋^{(𝑛−1)(𝑛−2)}Γ_𝑛−1(𝑛+1,𝜏)Γ_𝑛−1(𝑛−1)Γ_𝑛−1()𝐶2𝑛,𝜏^𝜏(𝐼). (18)
Note that 𝜅₂(𝑊)=𝜆₁/𝜆_𝑛=(1−𝛾_𝑛)⁻¹; then, the distribution of 𝜅₂(𝑊)=(1−𝛾_𝑛)⁻¹ is given by 𝑓𝜅₂(𝑊)=𝜋^{𝑛(𝑛−1)}𝜎^2−𝑚𝑛Γ_𝑛(𝑛)Γ_𝑛(𝑚)−𝑛exp𝜎²𝜆₁×^∞_{𝑝=0𝜅∈𝒟𝑝}𝜆₁^{𝑚𝑛−𝑛+𝑝}𝜎^2𝑝𝑝!^∞_{𝑠=0𝜈∈𝒟𝑠𝜏∈𝒟𝑡}1()𝑠!𝑛−𝑚_𝜈×𝑔^𝜏_𝜈,𝜅𝐶_𝜏𝜅₂(𝑊)−1^{(𝑛−1)(𝑛+1)+𝑝+𝑠−1}×𝜅₂(𝑊)^{−(𝑛−1)(𝑛+1)−𝑝−𝑠+1}×𝑛²×Γ−1+𝑝+𝑠_𝑛−1(𝑛−1)𝜋^{(𝑛−1)(𝑛−2)}×Γ_𝑛−1(𝑛+1,𝜏)Γ_𝑛−1(𝑛−1)Γ_𝑛−1()𝐶2𝑛,𝜏^𝜏(𝐼). (19)

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.

Acknowledgments

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).