The beta-Jacobi matrix model, the CS decomposition, and generalized singular value problems

Alan Edelman and Brian D. Sutton

Found. Comput. Math. 8 (2008), no. 2, 259–285.

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. This new random matrix distribution in a sense extends Haar measure on the orthogonal (beta = 1) and unitary (beta = 2) groups to general beta and reveals some new connections between random matrix theory and the CS decomposition.