Some Characterizations of the Distribution of the Condition Number of a Complex Gaussian Matrix

Abstract The objective of this paper is to characterize the distribution of the condition number of a complex Gaussian matrix. Several new distributional properties of the distribution of the condition number of a complex Gaussian matrix are given. Based on such distributional properties, some characterizations of the distribution are given by truncated moment, order statistics and upper record values.


Introduction
Gaussian matrices appear in many elds of pure and applied sciences, such as, probability, statistics, multivariate statistics, linear algebra, actuarial science, physics, wireless communications, and polarimetric synthetic aperture radar (PolSAR). Many researchers have studied the distributions of the condition numbers of Gaussian and Wishart matrices, their properties and applications, for which the interested readers are referred to Demmel (1988), Edelman (1988Edelman ( , 1992, Chen  The organization of this paper is as follows. In Section 2, some preliminaries are given. In Section 3, several distributional properties of the distributions of the condition numbers of complex Gaussian matrices are presented. The characterizations of the distributions of the condition numbers of complex Gaussian matrices are provided in Section 4. In Section 5, we have computed the percentage points. In Section 6, the concluding remarks are given.

Some Preliminaries
In this section, we will rst provide some preliminaries on the notions of the condition number of a matrix which will be relevant for establishing the various results of our paper. Then, a brief discussion of complex Gaussian matrix will be presented.

. Condition Number of a Matrix
Condition number of a matrix is one of the widely used matrix features and has signi cant applications in many areas of numerical analysis, linear algebra, statistics, physics and engineering. The idea of the condition number of a matrix was rst introduced in 1948 by Turing (1948). Further developments continued with the contributions of Rice (1966), Skeel (1979), Geurts (1982), Demmel (1987), Edelman (1988), Trefethen and Bau (1997), Bottcher and Grudsky (1998), Trefethen and Viswanath (1998), and Higham (2002), among others. For recent work on condition numbers, see, for example, Hargreaves (2004), Xu and Zhang (2004), Acosta et al. (2006), and Ern and Guermond (2006), and references therein.

. . De nition
For a given non-singular matrix A ∈ n×n , and a matrix norm ||.||, a non-negative quantity κ(A) given by is de ned as the condition number of A and is denoted by Cond (A), see, for example, Datta (1995), and Leon (2006), among others. Since the condition number is de ned in terms of a particular matrix norm, many di erent matrix norms may be chosen. The most frequently used norm is the p-norms denoted as ||A||p , (p ≥ ). Some commonly used p-norms are maximum column-sum norm = ||A|| = max ≤j≤n m i= |a ij |, where a ij 's denote the components of the matrix A, maximum row-sum norm = ||A||∞ = max ≤i≤m n j= |a ij |, and the Hilbert or spectral norm de ned as ||A|| = maximumeigenvalueofA T A, where A T denotes the transpose of the matrix A, and if A is Hermitian or real and symmetric, then ||A|| = ρ(A), where ρ(A) denotes the spectral radius of A de ned as ρ(A) = max i |λ i |, where λ s are the eigenvalues of A. The condition number corresponding to the p-norm will be denoted κp(A). For a singular square matrix A n×n , that is, if A is not invertible, the condition number of A is in nite, that is, κ(A) = +∞.

. . Condition Number Based on Spectral Norm
Let ||.|| denote the spectral norm. Let A be a nonsingular n × n matrix. Let σ , σ , . . . , σn denote the n singular values of A. Note that if the columns of A are orthogonal, the condition number, κ (A), of A attains its minimum bound, that is, κ (A) ≥ , (see, for example, Golub and Van Loan (1996), among others). If κ (A) is large, then A is said to be an ill-conditioned matrix. If κ (A) is near unity, then A is said to be a well-conditioned matrix. Matrices with small condition numbers are said to be well-conditioned.

. Complex Gaussian Matrices
A matrix is called a random matrix if the entries of the matrix are random numbers from a speci ed distribution. If the distribution is Gaussian, then we call it a Gaussian random matrix. Consider an m × n rectangular random matrix X all of whose elements are independent standard normal random variables. We call such a random matrix (or its distribution) as a Gaussian matrix, and denote it by G(m, n).
A complex standard normally distributed random variable can be de ned as u + vi, i = √ − , where u and v are independent and identically distributed (i.i.d.) standard normal random variables. If all the components of a random matrix are independent complex standard normal random variables, we call it a complex Gaussian matrix; see, for example, Haagerup, and Thorbjørnsen (2003). LetX be such a complex Gaussian matrix of the form X + iX , where X and X are independent and have the distribution G(m, n). LetG(m, n) denote the distribution ofX. LetX T denote the transpose of the n × n random matrixX. Then, the n × n random matrix Wn×n =XX T is said to be a Wishart matrix or to have the Wishart distribution with parameters (n, n).
Further, let λmax = λ > λ · · · > λn = λ min > and σ > σ · · · > σn > denote the distinct eigenvalues of the matrix Wn×n =XX T and singular values ofX, respectively. Note that the squares of the singular values ofX are the eigenvalues of the Wishart matrix Wn×n =XX T . Then, the random quantity κ (X) = λ λn = λmax λ min = σ σn , the square root of the ratio of largest to smallest eigenvalues of the Wishart matrix Wn×n =XX T , is de ned as the -norm (or the standard) condition number ofX, see, for example, Edelman (1988).

Distributional Properties of the Condition Numbers of Complex Gaussian Matrices
In this section, we discuss the properties of the distribution of the condition numbers of complex Gaussian matrices. For the sake of simplicity and without loss of generality, we shall consider the explicit expressions of the probability density function (pdf) of the exact distribution of the distribution of the condition number, κ, of a complex Gaussian matrix,X, having the distributionG( , n).

. Probability Density Function
Edelman (1988) introduced a distribution for a continuous random variable X to study the distributionG( , n) of the condition number,κ, of a complex Gaussian matrix,X, with the probability density function (pdf), f (x), given by By using the substitution t + = u , the corresponding cumulative distribution function (cdf) is easily derived as follows: where Bx(a, b) = x u a− ( − u) b− du denotes the incomplete beta function, see, for example, Abramowitz and Stegun (1970), and Gradshteyn and Ryzhik (1990). For some selected values of the parameter, n, the graphs of f (x) and F(x) are given in Figures 1 and 2 respectively. The e ects of the parameters are obvious from these gures, that is, the distribution appears to be unimodal and right skewed.

. Survival and Hazard Functions
The survival and hazard functions are respectively given by and where x > , n ≥ . The possible shapes of h(x) are provided for some selected values of the parameter, n, in the Figure 3. The e ects of the parameters can easily be seen from these graphs. The upside down bathtub shape behaviors of h(x) are evident from the gure.

. Moment
The jth moment, E(X j ), where j > is an integer is given by from which on substituting x + = u , we obtain , the rst moment is given by For some values of the parameter, n ≥ , the rst moment, E(X), is tabulated from Eq. (3.6), in Table 1.

. Entropy
The entropy measure of a random variable X is a measure of variation of uncertainty and has been used in many elds such as physics, engineering and economics, among others. According to Shannon (1948), the Thus, the entropy measure of the random variable X having the pdf f (x) as in Eq. (3.1) is given by where n ≥ , and x > . It is obvious from the above that we cannot obtain a closed form expression for the Shannon entropy. For some values of the parameter n ≥ , using the Maple software, the Shannon entropy is tabulated in Table 2, and the corresponding graph is sketched in Figure 4 below. From Table 2 and Figure 4, it is observed that the Shannon entropy is a monotonic decreasing and concave up function of n.

Characterization Results
In this section, we give our proposed characterizations of the distribution of condition number of a complex Gaussian matrix by truncated moment, order statistics and upper record values.
Many authors and researchers have studied the characterizations of probability distributions, among them Galambos and Kotz (1978), Kotz and Shanbhag (1980), and Nagaraja (2006), are notable. Since the truncated distributions arise in practical statistics where the ability of record observations is limited to a given threshold or within a speci ed range, there has been a great interest, in recent years, in the characterizations of probability distributions by truncated moments, that is, by considering a product of reverse hazard rate and another function of the truncated point, see, for example, Ahsanullah et al. (2015, 2016), and Ahsanullah and Shakil (2018).
Recently, many researchers have also used the idea of truncated moment method in the studies of the characterizations of the distributions of the condition numbers (DCN) of random matrices, see, for example, Shakil and Ahsanullah (2016). It appears from the literature that no attention has been paid to the characterizations of the distribution of the condition number of a complex Gaussian matrix. Therefore, motivated by the importance of the characterizations of probability distributions in practical problems, in this section we provide some new characterization results of the distribution of the distribution of the condition number,κ, of a complex Gaussian matrix,X, having the distributionG( , n), with pdf (3.1) and cdf (3.2), by truncated moment, order statistics and upper record values.
For this, we will need the following assumption and lemmas. We assume that df (x) dx exits for all x, where < x < ∞. We also assume that E(X) exists, which is true in our case, since the pdf (3.1) has nite E(X) and, therefore, can easily be characterized using conditional expectation of the rst moment. Assumption 4.1. Suppose the random variable X is absolutely continuous with the cumulative distribution function F(x) and the probability density function f (x). We assume that γ = inf{x|F(x) > }, and δ = sup{x|F(x) < }, f (x) is a di erentiable function of x in the interval (γ, δ), and E(X) exists.
F(x) and g(x) is a continuous di erentiable function of x with the condition that Proof. For proof, see Shakil and Ahsanullah (2018).

Lemma 4.2. Under the Assumption 4.1, if E(X|X
where c is a constant determined by the condition ∫ δ γ f (x)dx = .

. Characterization by Truncated Moment
Theorem 4.1. Suppose the random variable X is absolutely continuous with the cumulative distribution function F(x) and the probability density function f (x). We assume that = inf{x|F(x) > }, ∞ = sup{x|F(x) < },  Thus, we have This completes the proof of the necessary part. Su cient Part: Conversely, suppose that where P(x) is given by Eq. (4.3). Then, di erentiating both sides of the above equation with respect to x, and noting that d (Bx(a,b)) Since, by Lemma 4.1, On integrating both sides of the above equation with respect to x, we obtain ln f (x) = ln(c) + ( n − ) ln(x) + ln(x − ) − n ln(x + ), or, where c is a constant to be determined. Now, integrating the Eq. (4.4) with respect to x from to ∞, using the boundary condition ∞ f (x)dx = , we obtain c = Γ( n) Γ(n)Γ(n− ) . Then, we have This completes the proof of Theorem 4.1.

Theorem 4.2.
Suppose the random variable X is absolutely continuous with the cumulative distribution function F(x) and the probability density function f (x). We assume that = inf{x|F(x) > }, ∞ = sup{x|F(x) < }, and E(X) is nite for all x, where < x < ∞. Then, if and only if where E(x) denotes the rst moment given by Eq. (3.6), and P(x) is given by Eq. (4.3).
Proof. Necessary Part: Consider the pdf (3.1), that is, , we have where E(x) denotes the rst moment given by Eq. (3.6), and P(x) is given by Eq. (4.3). It follows from (4.5) that This completes the proof of the necessary part.

Su cient Part: Conversely, suppose that
Then, di erentiating both sides of the above equation with respect to x, and simplifying, we easily obtain Since, by Lemma 4.2, we have On integrating both sides of the above Eq. (4.8) with respect to x, we obtain ln f (x) = ln(c) + ( n − ) ln(x) + ln(x − ) − n ln(x + ), or, where c is a constant to be determined. Now, integrating the Eq. (4.9) with respect to x from to ∞, and using the boundary condition ∞ f (x)dx = , we obtain c = Γ( n) Γ(n)Γ(n− ) . Then, we have This completes the proof of Theorem 4.2.

. Characterizations by Order Statistics
If X , X , ..., Xn be the n independent copies of the random variable X with absolutely continuous distribution function F(x) and pdf f (x), and if X ,n ≤ X ,n ≤ ... ≤ Xn,n be the corresponding order statistics, then it is known from Ahsanullah, et al. (2013), chapter 5, or Arnold, et al. (2005), chapter 2, that X j,n |X k,n = x, for ≤ k < j ≤ n, is distributed as the (j − k)th order statistics from (n − k) independent observations from the random variable V having the pdf and X i.,n |X k,n = x, ≤ i < k ≤ n, is distributed as ith order statistics from k independent observations from the random variable W having the pdf f W (w|x) where and T k,n = n−k (X k+ ,n + X k+ ,n + ... + Xn.n).

Theorem 4.3:
Suppose the random variable X satis es the Assumption 4.1 with γ = and δ = ∞, then where P(x) is given by Eq.

. Characterizationby Upper Record Values
For details on record values, see Ahsanullah (1995). Let X , X , ... be a sequence of independent and identically distributed absolutely continuous random variables with distribution function F(x) and pdf f (x). If Yn = max(X , X , ..., Xn) for n ≥ and Y j > Y j− , j > , then X j is called an upper record value of {Xn , n ≥ }.
The indices at which the upper records occur are given by the record times {U(n) > min(j|j > U(n + )., X j > X U(n− ) , .n > )} and U( ) = . Let the nth upper record value be denoted by X(n) = X U(n) .
where E(x) denotes the rst moment given by Eq. (3.6), and P(x) is given by Eq. (4.3).

Percentiles
For any statistical applications, it is important to know the percentage points of a given distribution. For example, one may be interested in knowing the median (50%), 25%, or 75% quartiles, or 5%, 90%, 95%, or 99% con dence levels for other applications, to assess the statistical signi cance of an observation whose distribution is known. The pth percentile or the quantile of order p, for any < p < , of our distribution with the pdf f X (x) as in Eq.  Table 3 below.

Concluding Remarks
The condition numbers of Gaussian matrices occur in many branches of pure and applied sciences, and engineering. Many researchers have studied the distributions of condition numbers of Gaussian matrices. In this paper, several new distributional properties of the distribution of the condition number of a complex Gaussian matrix are presented. Based on the distributional properties, we have established some new characterization results. For some values of the parameter, n ≥ , the rst moment, E(X), and the Shannon entropy are numerically computed. Since, for any statistical applications, it is important to know the percentage points of a given distribution, we also have computed the percentiles of the distribution. It is hoped that the ndings of this paper will be quite useful for researchers in various elds of pure and applied sciences.