THE BETA-MANOVA ENSEMBLE WITH GENERAL COVARIANCE

Abstract

We consider adding arbitrary covariance to the β-Jacobi random matrix model. We recall that for β = 1 the Jacobi random matrix model may be thought of as the eigenvalues, λi, of YtY(XTX + YtY)-1 where X and Y are matrices whose elements are i.i.d. standard normals. Equivalently we can take the generalized cosine singular values of (Y, X), ci, and use [Formula: see text]. When β = 1 we add covariance by considering YtY(YtY + ΩXtXΩ)-1, for a positive definite diagonal matrix Ω. Equivalently, and preferably, we consider the generalized singular value decomposition (gsvd) of (Y, XΩ). We refer to Ω = I as the Jacobi case and the general Ω case as the MANOVA case. In this paper, we provide a matrix model for the general β-MANOVA ensemble. In particular, we provide an algorithm for the numerical sampling of eigenvalues or generalized cosine singular values. The β-MANOVA algorithm uses the β-Wishart algorithm of Forrester and Dubbs–Edelman–Koev–Venkataramana as a subroutine, perhaps making β-MANOVA the first "second-order" continuous-β random matrix algorithm. Our proofs make use of a conjecture of MacDonald (proven by Baker and Forrester), a theorem of Kaneko, and many identities from Forrester's Log-Gases and Random Matrices. We supply numerical evidence that our theorems are correct.