Smallest eigenvalue distributions for two classes of β-Jacobi ensembles

We compute the exact and limiting smallest eigenvalue distributions for two classes of β-Jacobi ensembles not covered by previous studies. In the general β case, these distributions are given by multivariate hypergeometric 2F1 2/β functions, whose behavior can be analyzed asymptotically for special values of β which include β ∈ 2N+ as well as for β = 1. Interest in these objects stems from their connections (in the β = 1, 2 cases) to principal submatrices of Haar-distributed (orthogonal, unitary) matrices appearing in randomized, communication-optimal, fast, and stable algorithms for eigenvalue computations [8], [4].