The Beta-MANOVA Ensemble with General Covariance

We find the joint generalized singular value distribution and largest generalized singular value distributions of the β-MANOVA ensemble with positive diagonal covariance, which is general. This has been done for the continuous β > 0 case for identity covariance (in eigenvalue form), and by setting the covariance to I in our model we get another version. For the diagonal covariance case, it has only been done for β = 1, 2, 4 cases (real, complex, and quaternion matrix entries). This is in a way the first second-order βensemble, since the sampler for the generalized singular values of the β-MANOVA with diagonal covariance calls the sampler for the eigenvalues of the β-Wishart with diagonal covariance of Forrester and Dubbs-Edelman-Koev-Venkataramana. We use a conjecture of MacDonald proven by Baker and Forrester concerning an integral of a hypergeometric function and a theorem of Kaneko concerning an integral of Jack polynomials to derive our generalized singular value distributions. In addition we use many identities from Forrester’s Log-Gases and Random Matrices. We supply numerical evidence that our theorems are correct.