A CLASS OF INTEGRAL IDENTITIES WITH HERMITIAN MATRIX ...

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 134, Number 11, November 2006, Pages 3329–3341 S 0002-9939(06)08602-3 Article electronically published on May 12, 2006

A CLASS OF INTEGRAL IDENTITIES WITH HERMITIAN MATRIX ARGUMENT ´ DAYA K. NAGAR, ARJUN K. GUPTA, AND LUZ ESTELA SANCHEZ (Communicated by Richard A. Davis)

Abstract. The gamma, beta and Dirichlet functions have been generalized in several ways by Ingham, Siegel, Bellman and Olkin. These authors defined them as integrals having the integrand as a scalar function of real symmetric matrix. In this article, we have defined and studied these functions when the integrand is a scalar function of Hermitian matrix.

1. Introduction Several generalizations of Euler’s gamma function are available in the scientific literature. The multivariate gamma function which is frequently used in multivariate statistical analysis is defined by (Ingham [4], Siegel [10])
(1.1) Γm (a) = etr(−X) det(X)a−(m+1)/2 dX, X>0

where etr(A) = exp(tr(A)), det(A) = determinant of A, and the integration is carried out over m × m symmetric positive definite matrices. By evaluating the above integral it is easy to see that   m  j−1 m−1 m(m−1)/4 . Γ a− (1.2) Γm (a) = π , Re(a) > 2 2 j=1 The multivariate generalization of the beta function is given by
(1.3) det(X)a−(m+1)/2 det(Im − X)b−(m+1)/2 dX Bm (a, b) = 0 (m−1)/2. Siegel [10] established the identity
det(Y )a−(m+1)/2 det(Im + Y )−(a+b) dY (1.4) Bm (a, b) = Y >0

Received by the editors June 10, 2003 and, in revised form, November 5, 2004 and June 1, 2005. 2000 Mathematics Subject Classification. Primary 33E99; Secondary 62H99. Key words and phrases. Beta function, Dirichlet function, gamma function, Liouville integral, matrix variate, transformation. The first and third authors were supported by the Comit´e para el Desarrollo de la Investigaci´ on, Universidad de Antioquia research grant no. IN387CE. c
2006 American Mathematical Society Reverts to public domain 28 years from publication

3329

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

3330

´ DAYA K. NAGAR, ARJUN K. GUPTA, AND LUZ ESTELA SANCHEZ

which can be derived from (1.3) by using the matrix transformation X = (Im + Y )−1 Y . The multivariate gamma function defined above arises naturally in the probability density function (p.d.f.) of the Wishart matrix. If W ∼ Wm (n, Σ), n > m − 1, Σ > 0, then its p.d.f. is given by    n −1 Σ−1 W nm/2 n/2 2 det(Σ) Γm etr − det(W )(n−m−1)/2 , W > 0. 2 2 Further, if W1 ∼ Wm (n1 , Im ) and W2 ∼ Wm (n2 , Im ) are independent Wishart matrices, then the p.d.f.’s of X = (W1 + W2 )−1/2 W1 (W1 + W2 )−1/2 and Y = −1/2 −1/2 W2 W1 W2 are given by   n n −1 1 2 , Bm det(X)n1 /2−(m+1)/2 det(Im − X)n2 /2−(m+1)/2 , 0 < X < Im , 2 2 and   n n −1 1 2 Bm , det(Y )n1 /2−(m+1)/2 det(Im + Y )−(n1 +n2 )/2 , Y > 0, 2 2 respectively. Thus multivariate gamma and multivariate beta functions frequently occur in the densities of random matrices (e.g., see Gupta and Nagar [3]) and hence play a pivotal role in multivariate statistical analysis. Let A = (aij ) be an m × m symmetric matrix. Define A(α) = (aij ), 1 ≤ i, j ≤ α. Bellman [1] has defined a multivariate gamma function as
etr(−X) det(X)am −(m+1)/2 m dX, Γm (a1 , . . . , am ) = kα−1 X>0 α=2 det(X(α) ) where aj = km−j+1 + · · · + km and Re(aj ) > (j − 1)/2, j = 1, . . . , m. Using an inductive argument, Bellman [1] showed that   m  j−1 m−1 . Γ aj − Γm (a1 , . . . , am ) = π m(m−1)/4 , Re(aj ) > 2 2 j=1 Olkin [7], using matrix transformation, evaluated the above integral and established similar results for the beta function. He also derived a matrix variate version of the Liouville integral. It may be noted here that integrands in (1.1), (1.3) and (1.4) are real-valued non-negative functions of symmetric matrices and have been evaluated over the space of positive definite matrices. When these functions (or their modified forms) have a Hermitian matrix as an argument and are integrated over Hermitian positive definite matrices, complex multivariate gamma and complex multivariate beta functions are defined. These functions occur in the p.d.f. of complex Wishart and complex beta matrices and play an important role in complex multivariate statistical analysis. In this article we study Bellman’s gamma function and Olkin’s beta function in the case of Hermitian positive definite matrices. We also study the Liouville integral and its generalizations to the complex case. In Section 2 and Section 3 the generalized complex multivariate gamma and generalized complex multivariate beta functions are defined, and several properties have been studied. In Section 4, the complex multivariate Dirichlet function is studied, and the relationship between the Dirichlet function and the complex multivariate gamma function is established. Finally, in Section 5, using results of Section 2 and Section 3, we define generalized complex matrix variate gamma and

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

INTEGRAL IDENTITIES WITH HERMITIAN MATRIX ARGUMENT

3331

generalized complex matrix variate beta distributions. The distributions so defined are closely related to the complex matrix variate gamma and the complex matrix variate beta distributions.

2. Generalized complex multivariate gamma function In this section we will define generalized complex multivariate gamma function. We first state the following notations and results (Khatri [5], Srivastava [13]) that will be utilized in this and subsequent sections. Let A = (aij ) be an m × m matrix of complex numbers. Then, A
denotes the transpose of A; A¯ denotes conjugate of A; AH denotes conjugate transpose of A; tr(A) = a11 + · · · + amm ; etr(A) = exp(tr(A)); det(A) = determinant of A; A = AH > 0 means that A is Hermitian positive definite and A1/2 denotes the unique Hermitian positive definite square root of A = AH > 0. The submatrices A(α) and A(α) , 1 ≤ α ≤ m, of the matrix A are defined as A(α) = (aij ), 1 ≤ i, j ≤ α, and A(α) = (aij ), α ≤ i, j ≤ m, respectively. Lemma 2.1 (Khatri [5]). Let Z and W be m × m Hermitian positive definite matrices. If Z = GW GH , where G (m × m) is a complex nonsingular matrix, then J(Z → W ) = det(GGH )m . Lemma 2.2 (Goodman [2]). Let W be a Hermitian positive definite matrix and W = T T H , where T is an m × m complex triangular matrix with positive diagonal 2(m−i)+1 elements. Then, (i) J(W → T ) = 2m m if T is lower triangular, and i=1 tii m 2i−1 m (ii) J(W → T ) = 2 t if T is upper triangular. i=1 ii Lemma 2.3 (Khatri [5]). Let X = (xij ), Y = (yij ) and T = (tij ) be m×m complex triangular matrices with positive diagonal elements and Y = T X. If T and X are m 2(m−i)+1 m lower triangular, then J(Y → T ) = i=1 xii and J(Y → X) = i=1 t2i−1 . m 2i−1 ii Further, if T and X are upper triangular, then J(Y → T ) = i=1 xii and m 2(m−i)+1 J(Y → X) = i=1 tii . Lemma 2.4. Let W and A be m × m Hermitian positive definite matrices and W = T AT H , where T is an m×m complex triangular matrix with positive diagonal m 2(m−i)+1 m−1 elements. Then, (i) J(W → T ) = 2m det(A) i=1 det(A(i) )2 i=1 tii if T m 2i−1 m m 2 is lower triangular and (ii) J(W → T ) = 2 det(A) i=2 det(A(i) ) if i=1 tii T is upper triangular. Proof. Write A = XX H , where X is a lower triangular matrix with positive diagonal elements, and let Y = T X. Then, using Lemma 2.2 and Lemma 2.3, m 2(m−i)+1 m 2(m−i)+1 J(W → T ) = J(W → Y )J(Y → T ) = (2m i=1 yii )( i=1 xii ) = m 4(m−i)+2 m 2(m−i)+1 4(m−i)+2 m {t xii }. The result follows by noting that i=1 xii = 2 i=1 ii (i) 2 det(A ) . The proof of (ii) is similar.
det(A) m−1 i=1 The complex multivariate gamma function is defined by
˜ (2.1) Γm (a) = etr(−X) det(X)a−m dX, X=X H >0

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

3332

´ DAYA K. NAGAR, ARJUN K. GUPTA, AND LUZ ESTELA SANCHEZ

where the integral is evaluated over m × m Hermitian positive definite matrices. Evaluation of the above integral yields

(2.2)

˜ m (a) = π m(m−1)/2 Γ

m 

Γ(a − j + 1), Re(a) > m − 1.

j=1

˜m (a, b), is defined by The complex multivariate beta function, denoted by B
(2.3)

det(X)a−m det(Im − X)b−m dX,

˜m (a, b) = B 0 m − 1. The complex multivariate beta function ˜m (a, b) can be expressed in terms of complex multivariate gamma functions as B

(2.4)

˜ ˜ ˜m (a, b) = Γm (a)Γm (b) = B ˜m (b, a). B ˜ m (a + b) Γ

Now we generalize the complex multivariate gamma function as follows. Definition 2.5. The generalized complex multivariate gamma function, denoted ˜ ∗m (a1 , . . . , am ), is defined as by Γ

(2.5)

˜ ∗m (a1 , . . . , am ) = Γ


X=X H >0

etr(−X) det(X)am −m m dX, kα−1 α=2 det(X(α) )

where aj = km−j+1 + · · · + km and Re(aj ) > j − 1, j = 1, . . . , m. Theorem 2.6. For Re(aj ) > j − 1, j = 1, . . . , m, ˜ ∗m (a1 , . . . , am ) = π m(m−1)/2 Γ

m 

Γ(aj − j + 1).

j=1

Proof. Let X = T T H , where T is an upper triangular matrix with positive diagonal elements, and partition X and T as  X=

X11 X21

X12 X22



 , T =

T11 0

T12 T22

 ,

where X11 and T11 are (α − 1) × (α − 1) matrices. m Now it is easy to see that m H , det(X(α) ) = i=α t2ii , det(X) = i=1 t2ii and tr(X) = tr(T T H ) = X(α) = T22 T22 √

m 2

m 2 2 −1 t2ij , 1 ≤ i < j ≤ m. From i j − 1, j = 1, . . . , m, and B = B H > 0,
˜ ∗m (a1 , . . . , am ) Γ etr(−BX) det(X)am −m m dX = , (2.8) m kα−1 (α) )kα X=X H >0 α=2 det(X(α) ) α=1 det(B where aj = km−j+1 + · · · + km . Proof. Since B = B H > 0, let B = CC H , where C = (cij ) is a lower triangular matrix. Substitute Λ = C H XC; then tr(BX) = tr(CC H X) = tr(C H XC) = tr(Λ), and J(X → Λ) = det(C)−2m = det(B)−m . Now partition C, X and Λ as       C11 0 X11 X12 Λ11 Λ12 C= , X= , Λ= , X21 X22 Λ21 Λ22 C21 C22

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

3334

´ DAYA K. NAGAR, ARJUN K. GUPTA, AND LUZ ESTELA SANCHEZ

where C11 , X11 and Λ11 are (α − 1) × (α − 1) matrices. Then, Λ22 = Λ(α) = m H H X22 C22 , and det(Λ(α) ) = det(C22 X22 C22 ) = det(X22 ) i=α c2ii = det(X(α) ) C22 m 2 i=α cii . Thus the left hand side of (2.8) reduces to
det(Λ)am −m etr(−Λ) det(B)−am m dΛ m m −2 kα−1 kα−1 Λ=ΛH >0 α=2 det(Λ(α) ) α=2 i=α (cii ) ˜ ∗ (a1 , . . . , am ) det(B)−am Γ = m m m . −2 kα−1 α=2 i=α (cii ) 2 (α−1) ).
Now the result follows by noting that m i=α cii = det(B)/det(B Theorem 2.9. For Re(bj ) > j − 1, j = 1, . . . , m, and B = B H > 0,
˜ ∗ (b1 , . . . , bm ) Γ etr(−BX) det(X)bm −m (2.9) dX = mm , m−1 nα (α) )nα+1 X=X H >0 α=1 det(B(α) ) α=1 det(X where bj = n1 + · · · + nj . Proof. Similar to the proof of Theorem 2.8.




Theorem 2.10. For Re(aj ) > j − 1, j = 1, . . . , m,  
m det(X)am −m etr(−X)(tr X)r ˜ ∗m (a1 , . . . , am ), m dX = aj Γ kα−1 X=X H >0 r α=2 det(X(α) ) j=1 where aj = km−j+1 + · · · + km with (a)0 = 1 and (a)r = a(a + 1) · · · (a + r − 1) = (a)r−1 (a + r − 1), r = 1, 2, . . . . Further, for Re(bj ) > j − 1, j = 1, . . . , m,  
m det(X)bm −m etr(−X)(tr X)r ˜ ∗m (b1 , . . . , bm ), Γ dX = b m−1 j (α) )nα+1 H det(X X=X >0 r α=1 j=1 where bj = n1 + · · · + nj . Proof. Setting B = tIm in (2.8) and differentiating the resulting identity r times with respect to t, we obtain
exp[−t tr(X)] det(X)am −m (tr X)r m dX kα−1 X=X H >0 α=2 det(X(α) )   m

˜ ∗ (a1 , . . . , am )t−( m α=1 αkα +r) . aj Γ = m j=1

r

Now, substituting t = 1 above we get the desired result. The proof of the second part follows similar steps.
3. Generalized complex multivariate beta function The generalized multivariate beta function is defined as follows. Definition 3.1. The generalized complex multivariate beta function, denoted by ∗ ˜m B (a1 , . . . , am ; b1 , . . . , bm ), is defined by ∗ ˜m B (3.1) (a1 , . . . , am ; b1 , . . . , bm )
det(X)am −m det(Im − X)bm −m dX m = , kα−1 det((I − X) nα−1 } m (α) ) 0 j − 1, j = 1, . . . , m.

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

INTEGRAL IDENTITIES WITH HERMITIAN MATRIX ARGUMENT

3335

Theorem 3.2. For Re(aj ) > j − 1, and Re(bj ) > j − 1, j = 1, . . . , m, ˜ ∗ (a1 , . . . , am )Γ ˜ ∗m (b1 , . . . , bm ) Γ ∗ ˜m B (a1 , . . . , am ; b1 , . . . , bm ) = m . ˜ ∗m (a1 + b1 , . . . , am + bm ) Γ Proof. We have, by Definition 2.5, ˜ ∗m (a1 , . . . , am )Γ ˜ ∗m (b1 , . . . , bm ) Γ

etr(−X − Y ) det(X)am −m det(Y )bm −m m = dX dY, kα−1 det(Y nα−1 } (α) ) X=X H >0 Y =Y H >0 α=2 {det(X(α) )

m

m where aj = i=m−j+1 ki , bj = i=m−j+1 ni , Re(aj ) > j − 1, and Re(bj ) > j − 1, j = 1, . . . , m. Now, let X + Y = T T H , where T is an upper trianguH −1 lar matrix with positive diagonal elements and W = T −1 X(T m ) 2 . Then X = H H T W T , Y = T (Im − W )T , det(Y(α) ) = det((Im − W )(α) ) i=α tii , det(X(α) ) =



2 det(W(α) ) m t2ii , and tr(X + Y ) = tr(T T H ) = m (t21ij + t22ij ) + m i=α i0,i=1,...,r

Substituting the Jacobian

r

i=1

det(Xi )

ai −m

 b−m  r r det Im − Xi dXi .

i=1

i=1

i=1

Xi = X, Wi = X −1/2 Xi X −1/2 , i = 1, . . . , r − 1, in (4.2) with

J(X1 , . . . , Xr−1 , Xr → W1 , . . . , Wr−1 , X) = det(X)(r−1)m , we get (4.3) φ(X) = det(X)a−m det(Im − X)b−m  ar −m r−1

r−1 r−1   ai −m × ··· det(Wi ) det Im − Wi dWi

i=1 0< r−1 i=1 Wi 0,i=1,...,r−1

i=1

i=1

˜m (a1 , . . . , ar−1 ; ar ). = det(X)a−m det(Im − X)b−m B Now from (4.3) and (2.3) we can write
˜m (a1 , . . . , ar ; b) = B (4.4)

φ(X) dX

0 j. Further, transforming U = T H A−1 T with the Jacobian 

−1 m m   2(i−1)+1 J(T → U ) = 2m det(A−1 ) det((A−1 )(i) )2 tii , i=2

i=1

we get the joint p.d.f. of A and U , etr(−A) det(A)ν1 +ν2 −m det(Im − U )ν1 −m det(U )ν2 −1 , ˜ m (ν1 )Γ ˜ m (ν2 ) m det(U(i) )2 Γ i=2 m 2i −1 )(i) )}. From (5.8), it is easy to see that since m i=1 tii = i=1 {det(U(i) )/ det((A A, i.e., A1 + A2 and U are independent, A ∼ CGm (ν1 + ν2 , Im ) and the distribution of U is given by (5.6). (ii) The proof is similar to part (i).
(5.8)

Theorem 5.5, in the real case, was derived by Olkin and Rubin [9]. References 1. R. Bellman, A generalization of some integral identities due to Ingham and Siegel, Duke Math. J., 23 (1956), 571–577. MR0081921 (18:468a) 2. N. R. Goodman, Statistical analysis based on a certain multivariate complex Gaussian distribution (an introduction), Ann. Math. Statist., 34 (1963), 152–177. MR0145618 (26:3148a) 3. A. K. Gupta and D. K. Nagar, Matrix Variate Distributions, Chapman & Hall/CRC, Boca Raton, 2000. MR1738933 (2001d:62055) 4. A. E. Ingham, An integral which occurs in statistics, Proc. Cambridge Philos. Soc., 29 (1933), 271-276. 5. C. G. Khatri, Classical statistical analysis based on a certain multivariate complex Gaussian distribution, Ann. Math. Statist., 36 (1965), 98–114. MR0192598 (33:823)

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

INTEGRAL IDENTITIES WITH HERMITIAN MATRIX ARGUMENT

3341

6. M. S. Klamkin, Extensions of Dirichlet’s multiple integral, SIAM J. Math. Anal., 2 (1971), 467–469. MR0286953 (44 #4160) 7. I. Olkin, A class of integral identities with matrix argument, Duke Math. J., 26 (1959), 207– 213. MR0101223 (21:36) 8. I. Olkin, Matrix extensions of Liouville-Dirichlet-type integrals, Linear Algebra Appl., 28 (1979), 155–160. MR0549430 (80i:26015) 9. I. Olkin and H. Rubin, Multivariate beta distributions and independence properties of the Wishart distribution, Ann. Math. Statist, 35 (1964), 261–269. Correction Ann. Math. Statist., 37 (1966), 297. MR0160297 (28:3511) ¨ 10. Carl Ludwig Siegel, Uber die analytische theorie der quadratischen formen, Ann. Math., 36 (1935), no. 3, 527–606. MR1503238 11. B. D. Sivazlian, The generalized Dirichlet’s multiple integral, SIAM Rev., 11 (1969), 285–288. MR0247014 (40:283) 12. B. D. Sivazlian, A class of multiple integrals, SIAM J. Math. Anal., 2 (1971), 72–75. MR0285680 (44:2898) 13. M. S. Srivastava, On the complex Wishart distribution, Ann. Math. Statist., 36 (1965), 313– 315. MR0172401 (30:2620) ´ticas, Universidad de Antioquia, Medell´ın, AA 1226, ColomDepartamento de Matema bia E-mail address: [email protected] Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, Ohio 43403-0221 E-mail address: [email protected] ´ticas, Universidad de Antioquia, Medell´ın, AA 1226, ColomDepartamento de Matema bia E-mail address: [email protected]

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use