Jacobians and Hypergeometric Functions in Complex Multivariate Analysis∗ T. Ratnarajah†‡
R. Vaillancourt†§
M. Alvo†¶
CRM-2927 July 2927
∗ Research
partially supported by NSERC and the CRM of the Universi´ e de Montr´ eal of Mathematics and Statistics, University of Ottawa, Ottawa, Ontario, Canada K1N 6N5 ‡
[email protected] §
[email protected] ¶
[email protected] † Department
Abstract In this paper, the Jacobians for complex matrix transformations are derived by means of the exterior product. The transformations are Cholesky factorization, eigendecomposition and orthonormal-triangular factorization, which frequently occur in complex multivariate analysis. As an example, using these Jacobians we derive the volume of Stiefel manifold and the complex noncentral Wishart density. Moreover, the multivariate densities are often represented by complex hypergeometric functions of matrix arguments, which can be expressed in terms of complex zonal polynomials. We give a method to compute these complex hypergeometric functions. Finally, applications of these random matrix theories to information theory and numerical analysis are mentioned. To appear the Canadian Applied Mathematics Quarterly Keywords and Phrases. Jacobians, complex random matrix, complex noncentral Wishart matrix, zonal polynomials, hypergeometric functions Mathematical Subject Classification. 62H10, 60E05, 94A15 R´ esum´ e On obtient le jacobien d’une transformation matricielle complexe au moyen du produit ext´erieur, telles les factorisations de Cholesky, de Schur et QR utilis´ees dans l’analyse complexe multidimensionnelle. On obtient aussi le volume d’une vari´et´e de Stiefel et la densit´e de Wishart noncentrale complexe. De plus, les densit´es multidimensionnelles sont souvent repr´esent´ees par des fonctions hyperg´eom´etriques de matrices qu’on peut repr´esenter au moyen de polynˆomes zonaux complexes. On donne une m´ethode pour calculer ces fonctions hyperg´eom´etriques. On mentionne des applications `a la th´eorie de l’information et aux m´ethodes num´eriques.
1
Introduction
It is known that calculating multivariate Jacobians is cumbersome and difficult. Moreover, the computation of the eigenvalue density function requires a method other than Jacobian calculations and that the densities could be calculated using differential forms defined on manifolds. In addition, the expression of the noncentral multivariate density function requires integration with respect to Haar measures on locally compact groups [6]. In this paper, we address these issues for complex multivariate densities. The Jacobians are derived by means of the exterior product approach. The complex noncentral Wishart density and the eigenvalue density of both central and noncentral complex Wishart matrices are often expressed in terms of complex hypergeometric functions, which are sums of zonal polynomials multiplied by hypergeometric coefficients [4], [7]. Since these complex hypergeometric functions are difficult to compute, we describe a method for computing them. This will enable us to compute the probabilities of multivariate density functions. It should be noted that the results presented in this paper have applications in information theory and numerical analysis, which are given in [8] and [9], respectively. In particular, we evaluate the capacity of multiple input, multiple output (MIMO) wireless communication systems and derive the condition number distribution of random matrices. In a wireless communication system, data is delivered from a transmitter to a receiver using radio waves or other electromagnetic waves. The waves, however, may be reflected off objects in the environment and scattered randomly while propagating from the transmitter to the receiver. Therefore, transmitted signals are attenuated and phase shifted during the transmission. This channel response can be modeled by the complex channel coefficients. Recently, researchers have investigated the use of multiple input, multiple output systems in response to the demand for higher bit rates in wireless communications. These studies show that MIMO systems increase the capacity significantly over single input, single output (SISO) systems. For example, a MIMO system achieves almost n more bits per hertz for every 3-dB increase in signal-to-noise ratio (SNR) compared to the SISO case, which only achieves one additional bit per hertz for every 3-dB increase in SNR, where n = min{nt , nr }, nt and nr are the number of inputs (or transmitters) and outputs (or receivers) of the wireless communication system. However, the channel coefficients from different transmitter antennas to a single receiver antenna can be correlated. This channel correlation degrades the capacity [10]. In Section 7, we show that the capacity of MIMO system can be evaluated using Wishart densities. This paper is organized as follows. In Section 2, we derive the Jacobians of complex matrix transformations. The Stiefel manifold is studied in Section 3 and the hypergeometric functions are described in Section 4. Then, in Section 5, the complex noncentral Wishart matrix is studied and, in Section 6, a method to evaluate hypergeometric functions is given. Finally, some applications are briefly mentioned in Section 7.
2
Exterior products and Jacobians
In order to transform density functions we need to compute the determinant of a matrix of partial derivatives (the so-called Jacobian). It is tedious to explicitly write down this determinant when dealing with many variables. To overcome this difficulty an equivalent approach based on the exterior product of differential forms was introduced in [3], [6]. This approach is outlined next for bivariate transformations. Consider the integral Z I= f (x1 , x2 )dx1 dx2 , (1) D
where D ⊂ R2 . By making the change of variables x1 = x1 (y1 , y2 )
and x2 = x2 (y1 , y2 ),
equation (1) becomes
Z I=
f (x(y)) det D0
∂xk ∂yl
dy1 dy2 ,
k, l = 1, 2,
(2)
where D0 denotes the image of D. An equivalent representation of equation (2) based on the exterior product of differential forms is given by Z ∂x1 ∂x1 ∂x2 ∂x2 I= f (x(y)) dy1 + dy2 ∧ dy1 + dy2 , (3) ∂y1 ∂y2 ∂y1 ∂y2 D0 where the exterior product of two differentials, dyk and dyl , satisfies 1
(i) dyk ∧ dyl = −dyl ∧ dyk
(skew-symmetry),
(ii) dyk ∧ dyk = 0. For an arbitrary n × m matrix X, the symbol (dX) denotes the exterior product of the mn elements of dX (dX) :=
m ^ n ^
dxkl .
l=1 k=1
For a symmetric m × m matrix X, the symbol (dX) denotes the exterior product of the m(m + 1)/2 distinct elements of dX ^ (dX) := dxkl . 1≤k≤l≤m
Similarly, if X is a skew-symmetric matrix (X = −X T ), then (dX) denotes the exterior product of the m(m − 1)/2 distinct elements of dX, and if X is an upper-triangular matrix, then ^ (dX) := dxkl . k≤l
Moreover, if X = Xr + iXc is a complex matrix, then (dX) = (dXr )(dXc ). The following theorem gives the Jacobian of a Hermitian transformation. Theorem 1. If X = BY B H , where X and Y are m × m Hermitian matrices and B is a nonsingular m × m matrix, then (dX) = (det B)2m (dY ). Proof. Note that the real and complex parts of Hermitian matrices are symmetric and skew-symmetric matrices, respectively. If X = Xr + iXc and Y = Yr + iYc , then Xr and Yr are symmetric matrices and Xc and Yc are skew-symmetric matrices. >From [6], we have (dXr ) = (det B)m+1 (dYr )
and (dXc ) = (det B)m−1 (dYc ).
Therefore, (dX) = (dXr )(dXc ) = (det B)2m (dYr )(dYc ) = (det B)2m (dY ). The Jacobian for the change of variables in the Cholesky factorization is given by the following theorem [2]. Here we derive this Jacobian based on the exterior product. Theorem 2. Let A be an m×m Hermitian positive definite matrix. The Cholesky factorization is given by A = T H T , where T is upper triangular with real and positive diagonal elements. Then we have (dA) = 2m
m Y
t2m−2k+1 (dT ). kk
k=1
Proof. Let A = (akl + ibkl ) and T = (tkl + iukl ). a11 a12 . . . a1m .. .. .. .. . . . . a1m a2m . . . amm
By considering the real part we have t211 t11 t12 . . . t11 t1m .. .. .. .. = . . . . . 2 t1m t11 ... . . . tmm + . . .
For the imaginary part
0 .. .
b12 .. .
... .. .
b1m
b2m
...
b1m 0 .. = .. . . 0 −u1m t11
t11 u12 .. .
... .. .
...
...
t11 u1m .. . . 0
Next, expressing the diagonal and the upper-diagonal elements of A in term of the elements of T and taking differentials we get the following for their real parts da11 = 2t11 dt11 da12 = t11 dt12 + . . . .. . damm = 2tmm dtmm + . . . , 2
and for their complex parts db12 = t11 du12 + . . . db13 = t11 du13 + . . . .. . dbm−1,m = tm−1,m−1 dum−1,m + . . . .
Note that we only retained the terms that contribute to the exterior product because the products of repeated differentials are zero. By taking the exterior product we get (dA) = (dAr )(dAc ) m m ^ ^ = dakl dbkl k≤l
k q + 1, then the series diverges for all X 6= 0, unless it terminates. Note that the series terminates when some of the numerators [aj ]κ in the series vanish. Furthermore, the complex zonal polynomial Cκ (Y ) is defined by
7
Cκ (Y ) = χ[κ] (1)χ[κ] (Y ), where χ[κ] (1) is the dimension of the representation [κ] of the symmetric group given by Qm i
