2.1.Types of Matrices and Non-Asymptotic Results 25 n x n (complex)Haar matrix,then U31=1 2 EU4]= n(m+1) 1 E[lU2Ukjl2]E[lU2JUi2] n(m+1) 1 E[lU2U2]= n2-1 1 EUURUU】=- n(n2-1) A way to generate a Haar matrix is the following:let H be an nxn stan- dard complex Gaussian matrix and let R be the upper triangular ma- trix obtained from the QR decomposition of H chosen such that all its diagonal entries are nonnegative.Then,as a consequence of Lemma 2.1, HR-1 is a Haar matrix [245]. 2.1.5 Unitarily Invariant Matrices Definition 2.6.A Hermitian random matrix W is called unitarily in- variant if the joint distribution of its entries equals that of VWVi for any unitary matrix V independent of W. Example 2.1.A Haar matrix is unitarily invariant. Example 2.2.A Gaussian Wigner matrix is unitarily invariant. Example 2.3.A central Wishart matrix W~Wm(n,I)is unitarily invariant. Lemma 2.6.(e.g [111])If W is unitarily invariant,then it can be decomposed as W=UAUt. with U a Haar matrix independent of the diagonal matrix A. Lemma 2.7.[110,111]If W is unitarily invariant and f()is a real continuous function defined on the real line,then f(W),given via the functional calculus,is also unitarily invariant
2.1. Types of Matrices and Non-Asymptotic Results 25 n × n (complex) Haar matrix, then E[|Uij | 2] = 1 n E[|Uij | 4] = 2 n(n + 1) E[|Uij | 2|Ukj| 2] = E[|Uij | 2|Ui| 2] = 1 n(n + 1) E[|Uij | 2|Uk| 2] = 1 n2 − 1 E[UijUkU∗ iU∗ kj ] = − 1 n(n2 − 1). A way to generate a Haar matrix is the following: let H be an n×n standard complex Gaussian matrix and let R be the upper triangular matrix obtained from the QR decomposition of H chosen such that all its diagonal entries are nonnegative. Then, as a consequence of Lemma 2.1, HR−1 is a Haar matrix [245]. 2.1.5 Unitarily Invariant Matrices Definition 2.6. A Hermitian random matrix W is called unitarily invariant if the joint distribution of its entries equals that of VWV† for any unitary matrix V independent of W. Example 2.1. A Haar matrix is unitarily invariant. Example 2.2. A Gaussian Wigner matrix is unitarily invariant. Example 2.3. A central Wishart matrix W ∼ Wm(n, I) is unitarily invariant. Lemma 2.6. (e.g [111]) If W is unitarily invariant, then it can be decomposed as W = UΛU† . with U a Haar matrix independent of the diagonal matrix Λ. Lemma 2.7. [110, 111] If W is unitarily invariant and f(·) is a real continuous function defined on the real line, then f(W), given via the functional calculus, is also unitarily invariant
26 Random Matrix Theory Definition 2.7.A rectangular random matrix H is called bi-unitarily invariant if the joint distribution of its entries equals that of UHVt for any unitary matrices U and V independent of H. Example 2.4.A standard Gaussian random matrix is bi-unitarily in- variant. Lemma 2.8.[111]If H is a bi-unitarily invariant square random ma- trix,then it admits a polar decomposition H=UC where U is a Haar matrix independent of the unitarily-invariant nonnegative definite ran- dom matrix C. In the case of a rectangular m x n matrix H,with m n,Lemma 2.8 also applies with C an n xn unitarily-invariant nonnegative definite random matrix and with U uniformly distributed over the manifold of complex m x n matrices such that UUt I. 2.1.6 Properties of Wishart Matrices In this subsection we collect a number of properties of central and non- central Wishart matrices and,in some cases,their inverses.We begin by considering the first and second order moments of a central Wishart matrix and its inverse Lemma 2.9.[164,96]For a central Wishart matrix W~Wm(n,I), E[tr{W】=mn E[trW2】=mn(m+n) E[tr2{W】=mn(mn+1): Lemma 2.10.[164,96](see also [133])For a central Wishart matrix W~Wm(n,I)with n>m, 卫w月=n”m (2.9) while,for n>m+1, E[tr{W-2】=m-m3-(n-m mn Ew-=nn(n--+nm+i） m-1
26 Random Matrix Theory Definition 2.7. A rectangular random matrix H is called bi-unitarily invariant if the joint distribution of its entries equals that of UHV† for any unitary matrices U and V independent of H. Example 2.4. A standard Gaussian random matrix is bi-unitarily invariant. Lemma 2.8. [111] If H is a bi-unitarily invariant square random matrix, then it admits a polar decomposition H = UC where U is a Haar matrix independent of the unitarily-invariant nonnegative definite random matrix C. In the case of a rectangular m × n matrix H, with m ≤ n, Lemma 2.8 also applies with C an n×n unitarily-invariant nonnegative definite random matrix and with U uniformly distributed over the manifold of complex m × n matrices such that UU† = I. 2.1.6 Properties of Wishart Matrices In this subsection we collect a number of properties of central and noncentral Wishart matrices and, in some cases, their inverses. We begin by considering the first and second order moments of a central Wishart matrix and its inverse. Lemma 2.9. [164, 96] For a central Wishart matrix W ∼ Wm(n, I), E[tr{W}] = mn E[tr{W2}] = mn (m + n) E[tr2{W}] = mn (mn + 1). Lemma 2.10. [164, 96](see also [133]) For a central Wishart matrix W ∼ Wm(n, I) with n>m, E tr W−1 = m n − m (2.9) while, for n>m + 1, E tr W−2 = m n (n − m)3 − (n − m) E tr2 W−1 = m n − m n (n − m)2 − 1 + m − 1 n − m + 1
2.1.Types of Matrices and Non-Asymptotic Results 27 For higher order moments of Wishart and generalized inverse Wishart matrices,see [96]. From Lemma 2.1,we can derive several formulas on the determinant and log-determinant of a Wishart matrix. Theorem 2.11.182,131]9 A central complex Wishart matrix W~ Wm(n,I),with n>m,satisfies aw时-9 (2.10) 0-0 and hence the moment-generating function of loge det W for >0 is w) I(n-E+S) 2.11) 0=0 r(m-) with I()denoting the Gamma function [97] r(a)= ta-le-idt Jo which,for integer arguments,satisfies I(n+1)=n!From(2.11), m-1 Elloge det W] ∑(n-0 (2.12) 0= m- Var[loge det W]= ∑(n-) 2.13 L=0 where (is Euler's digamma function [97],which for natural argu- ments can be expressed as m-11 (m)=(1)+ =(m-1)+ m-1 (2.14) 0=1 with -(1)=0.577215...the Euler-Mascheroni constant.The deriva- tive of ()in turn,can be expressed as (m+1)=0m）- m2 (2.15) Note that 182,131]derive the real counterpart of Theorem 2.11,from which the complex case follows immediately
2.1. Types of Matrices and Non-Asymptotic Results 27 For higher order moments of Wishart and generalized inverse Wishart matrices, see [96]. From Lemma 2.1, we can derive several formulas on the determinant and log-determinant of a Wishart matrix. Theorem 2.11. [182, 131]9 A central complex Wishart matrix W ∼ Wm(n, I), with n ≥ m, satisfies E detWk = m −1 =0 Γ(n − + k) Γ(n − ) (2.10) and hence the moment-generating function of loge detW for ζ ≥ 0 is E eζ loge detW = m −1 =0 Γ(n − + ζ) Γ(n − ) (2.11) with Γ(·) denoting the Gamma function [97] Γ(a) = ∞ 0 t a−1e−t dt which, for integer arguments, satisfies Γ(n + 1) = n! From (2.11), E[loge detW] = m −1 =0 ψ(n − ) (2.12) Var[loge detW] = m −1 =0 ψ˙(n − ) (2.13) where ψ(·) is Euler’s digamma function [97], which for natural arguments can be expressed as ψ(m) = ψ(1) + m −1 =1 1 = ψ(m − 1) + 1 m − 1 (2.14) with −ψ(1) = 0.577215... the Euler-Mascheroni constant. The derivative of ψ(·), in turn, can be expressed as ψ˙(m + 1) = ψ˙(m) − 1 m2 (2.15) 9 Note that [182, 131] derive the real counterpart of Theorem 2.11, from which the complex case follows immediately
28 Random Matrix Theory with()=若. If and are positive definite deterministic matrices and H is an n x n complex Gaussian matrix with independent zero-mean unit- variance entries,then W=HHT satisfies (using(2.10)) detW]=det(Φ (m-+k-1)月 (n--1)川 (2.16) = The generalization of(2.16)for rectangular H is derived in [165,219]. Analogous relationships for the non-central Wishart matrix are derived in[5. Theorem 2.12.[166]Let H be an n x m complex Gaussian matrix with zero-mean unit-variance entries and let W be a complex Wishart matrix W~Wn(p,I),with m≤n≤p.Then,for(∈(-l,l), m-1 Edet(HiWw-1H)的= I(m+p-n-S-e)I(n+S-E) E=0 I(n-e)I(m+p-n-e) m- Elog det(HTW-H)]= ∑((n-0-(m+p-n-O). =0 Additional results on quadratic functions of central and non-central Wishart matrices can be found in [141,142,144]and the references therein. Some results on the p.d.f.of complex pseudo-Wishart matrices1o and their corresponding eigenvalues can be found in [58,59,168]. Next,we turn our attention to the determinant and log-determinant of matrices that can be expressed as a multiple of the identity plus a Wishart matrix,a familiar form in the expressions of the channel capacity. 10W-HHt is a pseudo-Wishart matrix if H is a mxn Gaussian matrix and the correlation matrix of the columns of H has a rank strictly larger than n [244,267,94,58,59]
28 Random Matrix Theory with ψ˙(1) = π2 6 . If Σ and Φ are positive definite deterministic matrices and H is an n × n complex Gaussian matrix with independent zero-mean unitvariance entries, then W = ΣHΦH† satisfies (using (2.10)) E detWk = det(ΣΦ) k n −1 =0 (n − + k − 1)! (n − − 1)! (2.16) The generalization of (2.16) for rectangular H is derived in [165, 219]. Analogous relationships for the non-central Wishart matrix are derived in [5]. Theorem 2.12. [166] Let H be an n × m complex Gaussian matrix with zero-mean unit-variance entries and let W be a complex Wishart matrix W ∼ Wn(p, I), with m ≤ n ≤ p. Then, for ζ ∈ (−1, 1), E[det(H† W−1H) ζ ] = m −1 =0 Γ(m + p − n − ζ − ) Γ(n + ζ − ) Γ(n − ) Γ(m + p − n − ) E[log det(H† W−1H)] = m −1 =0 (ψ(n − ) − ψ(m + p − n − )). Additional results on quadratic functions of central and non-central Wishart matrices can be found in [141, 142, 144] and the references therein. Some results on the p.d.f. of complex pseudo-Wishart matrices10 and their corresponding eigenvalues can be found in [58, 59, 168]. Next, we turn our attention to the determinant and log-determinant of matrices that can be expressed as a multiple of the identity plus a Wishart matrix, a familiar form in the expressions of the channel capacity. 10W = HH† is a pseudo-Wishart matrix if H is a m×n Gaussian matrix and the correlation matrix of the columns of H has a rank strictly larger than n [244, 267, 94, 58, 59]
2.1.Types of Matrices and Non-Asymptotic Results 29 Theorem 2.13.A complex Wishart matrix W~Wm(n,I),with n m,satisfies m Edet(I+yW)】=】 m n! (2.17) i=0 Theorem 2.14.[38,299]Let W be a central Wishart matrix W~ Wm(n,I)and let t minfn,m}and r maxfn,m}.The moment- generating function of loge det(I+W)is [o.da(w) detG() Π=1r- (2.18) with G(S)a t x t Hankel matrix whose (i,k)th entry is Gi.ke= (1+7X)s xd-le-Xd 人 y-d(d-1)1 T(-)sin(π(d-1+G)T(1+d+) B(d,1+d+,) (61-d-》 (2.19) with 1F1()the confluent hypergeometric function [97]and with d r-t+i+k+1. For a non-central Wishart matrix with covariance matrix equal to the identity,a series expression for Elog det(I+W)]has been com- puted in [3]while the moment-generating function(2.18)has been com- puted in [134]in terms of the integral of hypergeometric functions. For a central Wishart matrix WWm(n,D)where is posi- tive definite with distinct eigenvalues,the moment-generating function (2.18)has been computed in [234]and [135].11 Theorem 2.15.[192]If H is an m x m zero-mean unit-variance com- plex Gaussian matrix and and Y are positive definite matrices having 11 Reference [234]evaluates (2.18)in terms of Gamma functions for m>n while reference [135]evaluates it for arbitrary m and n,in terms of confluent hypergeometric functions of the second kind 97
2.1. Types of Matrices and Non-Asymptotic Results 29 Theorem 2.13. A complex Wishart matrix W ∼ Wm(n, I), with n ≥ m, satisfies E[det(I + γW)] = m i=0 m i n! (n − i)! γi . (2.17) Theorem 2.14. [38, 299] Let W be a central Wishart matrix W ∼ Wm(n, I) and let t = min{n, m} and r = max{n, m}. The momentgenerating function of loge det(I + γW) is E eζ loge det(I+γW) = detG(ζ) t i=1(r − i)! (2.18) with G(ζ) a t × t Hankel matrix whose (i, k)th entry is Gi,k = ∞ 0 (1 + γλ) ζ λd−1e−λdλ = π Γ(−ζ) sin(π(d − 1 + ζ)) γ−d (d − 1)! Γ(1 + d + ζ) 1F1 d, 1 + d + ζ, 1 γ − γζ Γ(−ζ) Γ(1 − d − ζ) 1F1 −ζ, 1 − d − ζ, 1 γ (2.19) with 1F1(·) the confluent hypergeometric function [97] and with d = r − t + i + k + 1. For a non-central Wishart matrix with covariance matrix equal to the identity, a series expression for E[log det(I + γW)] has been computed in [3] while the moment-generating function (2.18) has been computed in [134] in terms of the integral of hypergeometric functions. For a central Wishart matrix W ∼ Wm(n, Σ) where Σ is positive definite with distinct eigenvalues, the moment-generating function (2.18) has been computed in [234] and [135].11 Theorem 2.15. [192] If H is an m × m zero-mean unit-variance complex Gaussian matrix and Σ and Υ are positive definite matrices having 11 Reference [234] evaluates (2.18) in terms of Gamma functions for m>n while reference [135] evaluates it for arbitrary m and n, in terms of confluent hypergeometric functions of the second kind [97].