The beta-Jacobi matrix model, the CS decomposition, and generalized singular value problems
We provide a solution to the beta-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 beta-Jacobi matrix model. The Jacobi ensemble on R(n), parametrized by beta > 0, a > -1,and b > -1, is the probability distribution whose density is proportional to Pi(i)lambda((beta/2)(a+1)-1)(i) (1-lambda(i))((beta/2)(b+1)-1)Pi(i  0. Observing a connection between Haar measure on the orthogonal (resp., unitary) group and pairs of real (resp., complex) Gaussian matrices, we find a direct connection between multivariate analysis of variance (MANOVA) and the new matrix model.
cs decomposition random matrix jacobi ensemble generalized singular value decomposition