The Beta-Jacobi Matrix Model, the CS Decomposition, and Generalized Singular Value Problems

Abstract. We provide a solution to the β-Jacobi matrix model problem posed by Dumitriu and the first author. The random matrix distribution introduced here, called a matrix model, is related to the model of Killip and Nenciu, but the development is quite different. We start by introducing a new matrix decomposition and an algorithm for computing this decomposition. Then we run the algorithm on a Haar-distributed random matrix to produce the β-Jacobi matrix model. The Jacobi ensemble on R, parameterized by β > 0, a > −1, and b > −1, is the probability distribution whose density is proportional