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Eigenvalue Distributions of Beta-Wishart Matrices Eigenvalue distributions of beta-Wishart matrices The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Edelman, Alan, and Plamen Koev. “Eigenvalue Distributions of Beta- Wishart Matrices.” Random Matrices: Theory and Applications 03, no. 02 (April 2014): 1450009. As Published http://dx.doi.org/10.1142/S2010326314500099 Publisher World Scientific Pub Co Pte Lt Version Author's final manuscript Citable link http://hdl.handle.net/1721.1/116006 Terms of Use Creative Commons Attribution-Noncommercial-Share Alike Detailed Terms http://creativecommons.org/licenses/by-nc-sa/4.0/ SIAM J. MATRIX ANAL. APPL. c 2013 Society for Industrial and Applied Mathematics Vol. XX, No. X, pp. XX{XX EIGENVALUE DISTRIBUTIONS OF BETA-WISHART MATRICES∗ ALAN EDELMANy AND PLAMEN KOEVz Abstract. We derive explicit expressions for the distributions of the extreme eigenvalues of the Beta-Wishart random matrices in terms of the hypergeometric function of a matrix argument. These results generalize the classical results for the real (β = 1), complex (β = 2), and quaternion (β = 4) Wishart matrices to any β > 0. Key words. random matrix, Wishart distribution, eigenavalue, hypergeometric function of a matrix argument AMS subject classifications. 15A52, 60E05, 62H10, 65F15 DOI. XX.XXXX/SXXXXXXXXXXXXXXXX 1. Introduction. Recently, the classical real (β = 1), complex (β = 2), and quaternion (β = 4) Wishart random matrix ensembles were generalized to any β > 0 by what is now called the Beta{Wishart ensemble [2, 9]. In this paper we derive the explicit distributions for the extreme eigenvalues and the trace of this ensemble as series of Jack functions and, in particular, in terms of the hypergeometric function of matrix argument. These results generalize the classical expressions for the real and complex cases as series of zonal polynomials [17] and Schur functions [18], respectively. This paper is motivated by the importance of the real and complex Wishart matrices (and in particular their eigenvalue distributions) in multivariate statistical analysis. We refer to the classical text by Muirhead [17], as well as [8, 10, 18]. Recently, it is becoming increasingly apparent that the classical matrix ensembles (and perhaps all of random matrix theory) generalizes from the Dyson's three-fold way (β = 1; 2; 4) [5] to any β > 0 [6]. Recent examples include the Beta-Hermite [3], Beta{Laguerre [3], Beta{Jacobi [4, 7, 12, 14], and Beta{Wishart ensembles [2, 9]. Our new results are particularly convenient for practical evaluation using our algorithms for the hypergeometric function of matrix argument [13], see section 5. 2. Definitions and background. Since the eigenvalue distributions will be expressed as series of multivariate symmetric polynomials, we start with the relevant definitions. For an integer k ≥ 0 we say that κ = (κ1; κ2;:::) is a partition of k (denoted κ ` k) if κ1 ≥ κ2 ≥ · · · ≥ 0 are integers such that jκj ≡ κ1 + κ2 + ··· = k. Partitions with t equal parts are denoted as (a)t = (a; a; : : : ; a). For two partitions λ and κ we write λ ⊆ κ to indicate that λi ≤ κi for all i. ∗Received by the editors DATE; accepted for publication (in revised form) by EDITOR DATE; published electronically DATE. This research was supported by the National Science Foundation Grant DMS-1016086 and by the Woodward Fund for Applied Mathematics at San Jose State Uni- versity. The Woodward Fund is a gift from the estate of Mrs. Marie Woodward in memory of her son, Henry Teynham Woodward. He was an alumnus of the Mathematics Department at San Jose State University and worked with research groups at NASA Ames. http://www.siam.org/journals/simax/XX-X/XXXXX.html yDepartment of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139 ([email protected]). zDepartment of Mathematics, San Jose State University, 1 Washington Sq., San Jose, CA 95192 ([email protected]). 1 2 ALAN EDELMAN and PLAMEN KOEV For a partition κ = (κ1; κ2; : : : ; κm) and β > 0, the Generalized Pochhammer symbol of parameter β is defined as m κ Y Yi (i − 1)β (2.1) (a)(β) ≡ a − + j − 1 : κ 2 i=1 j=1 The multivariate Gamma function of parameter β is defined as m m(m−1)β Y (i − 1)β (m − 1)β (2.2) Γ(β)(c) = π 4 Γ c − for <(c) > : m 2 2 i=1 (β) For an m × m matrix X, the Jack functions Cκ (X) are symmetric homogeneous (β) polynomials in the eigenvalues x1; x2; : : : ; xm of X. The Cκ 's are indexed by parti- tions κ and form an orthogonal basis of the space multivariate symmetric polynomials. We refer to Stanley [19], Macdonald [16], and Forrester [8] for details and assume the reader has some familiarity with these functions. For integers p ≥ 0 and q ≥ 0, and an m × m complex symmetric matrices X, Y , the hypergeometric function of a matrix argument is defined as 1 (β) (β) (β) (β) (β) X X 1 (a1)κ ··· (ap)κ Cκ (X)Cκ (Y ) (2.3) pFq (a; b; X; Y ) ≡ · (β) (β) · (β) ; k! (b )κ ··· (b )κ Cκ (I) k=0 κ`k 1 q where (a) = (a1; a2; : : : ; ap) and b = (b1; b2; : : : ; bq). For one matrix argument X, (β) (β) pFq (a; b; X) ≡ pFq (a; b; X; I) Following [2], an m × m Beta{Wishart matrix with n (n ≥ m) degrees of freedom (β) and covariance matrix Σ (denoted A ∼ Wm (n; Σ)) has joint eigenvalue density nβ m − 2 (n−m+1)β jΣj Y −1 Y (β) (2.4) λ 2 jλ − λ jβ · F (− β Λ; Σ−1); (β) nβ i k j 0 0 2 Km ( 2 ) i=1 j<k where Λ = diag(λ1; λ2; : : : ; λm) are the eigenvalues of A, jΣj ≡ det Σ, and m! 2 ma Γ(β)(a)Γ(β) mβ (2.5) K(β)(a) ≡ · · m m 2 : m m(m−1)β β β m π 2 Γ 2 (β) 3. Identities involving pFq . We present several identities, including a new one, which we need in the next section. First, we incorporate the Vandermonde determinant into a new measure µ to prevent it from appearing in all the integrals that follow: Y β dµ(X) ≡ jxi − xjj dX: i<j Also, define (β) (β) Γm (b) Tm (a; b) ≡ (β) (β) ; Γm (a)Γm (b − a) m Γ( iβ ) (β) m! Y 2 cm ≡ m(m−1)β β ; π 4 i=1 Γ( 2 ) m − 1 (3.1) p ≡ β + 1: 2 EIGENVALUES OF BETA{WISHART MATRICES 3 Then, for a nonnegative integer r, (β) tr X (β) 0F0 (X; I + Y ) = e 0F0 (X; Y );(3.2) (β) r 1F0 (−r; X) = jI − Xj ;(3.3) (β) (β) −1 −b 2F1 (a; b; c; X) = 2F1 c − a; b; c; −X(I − X) · jI − Xj ;(3.4) (β) (β) (β) Γm (c)Γm (c − a + r) (3.5) 2F1 (a; −r; c; X) = (β) (β) Γm (c − a)Γm (c + r) (β) m−1 × 2F1 a; −r; a − r + 1 + 2 β − c; I − X ; (β) Z (β) Tm (a; b) (β) a−p b−a−p (3.6) 1F1 (a; b; Y ) = (β) 0F0 (X; Y )jXj jI − Xj dµ(X); cm [0;1]m Z (β) 1 −tr (X) (β) a−p (3.7) 1F0 (a; Y ) = (β) (β) e 0F0 (X; Y )jXj dµ(X); cm Γm (a) m R+ a+p (β) −1 jY j (3.8) Cκ (Y ) = (β) (β) (β) cm Γm (a + p) · (a + p)κ Z (β) a (β) × 0F0 (−X; Y )jXj Cκ (X)dµ(X); + Rm (β) mr (β) −1 (β) Γm (p + r) r X X Cκ (−X ) (3.9) 2F0 (p; −r; X) = (β) j − Xj : Γm (p) k! k=0 κ`k; κ1≤r The identities (3.2) and (3.8) were conjectured by Macdonald [15] and proven by Forrester and Baker [1, section 6]. The identities (3.3), (3.4), and (3.5) are due to Forrester: see (13.4) and Propo- sitions 13.1.6 and 13.1.7 in [8], respectively. Note that (3.3) was established for ar- guments X whose eigenvalues are in [0; 1]. When r ≥ 0 is an integer, however, (3.3) is true for any X|we have polynomials on both sides,1 which must be identical everywhere. The integrals (3.6) and (3.7) are due to Macdonald [15, (6.20), (6.21)]. For com- pleteness, we repeat his argument here. The integral (3.6) follows directly from Kadell's extension of Selberg's integral [11, Theorem 1] Z (β) (β) (β) Cκ (X) a−p b−a−p cm (a)κ (3.10) (β) jXj jI − Xj dµ(X) = (β) · (β) [0;1]m Cκ (I) Tm (a; b) (b)κ 1 (β) by multiplying both sides by jκj! Cκ (Y ) and summing over all κ. To deduce (3.7), we change variables yi = Nxi; i = 1; 2; : : : ; m, in (3.10), where N ≡ b − p, to obtain m Z (β) (β) (β) Cκ (Y ) a−p Y yi N ma+jκj cm (a)κ (β) jY j 1 − N dµ(Y ) = N · (β) · (β) m C (I) T (a; N + p) (N + p) [0;N] κ i=1 m κ (β) ma+jκj (β) (β) (β) Γm (N + p − a)N = Γm (a)cm (a)κ (β) (β) : Γm (N + p)(N + p)κ 1 (β) In the series for 1F0 on the left, if κ has any part greater than r or has more than m parts, (β) (β) then (−r)κ = 0 or Cκ (X) = 0, respectively.
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