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 \({\Bbb R}^{n}\) , parametrized by β > 0, a > -1,and b > -1, is the probability distribution whose density is proportional to \(\prod_{i}\lambda_{i}^{({\beta}/{2})(a+1)-1}(1-\lambda_{i})^{({\beta}/{2})(b+1)-1}\prod_{i<j}|\lambda_{i}-\lambda_{j}|^{\beta}\) . The matrix model introduced in this paper is a probability distribution on structured orthogonal matrices. If J is a random matrix drawnfrom this distribution, then a CS decomposition can be taken,$$J=\left[\begin{array}{@{}cc@{}} U_{1}\\ & U_{2}\end{array}\right]\left[\begin{array}{@{}rc@{}} C & S\\-S & C\end{array}\right]\left[\begin{array}{@{}cc@{}} V_{1}\\ & V_{2}\end{array}\right]^{T}$$, in which C and S are diagonal matrices with entries in [0,1]. J is designed so that the diagonal entries of C, squared, follow the law of the Jacobi ensemble. When β = 1 (resp., β = 2), the matrix model is derived by running a numerically inspired algorithm on a Haar-distributed random matrix from the orthogonal (resp., unitary) group. Hence, the matrix model generalizes certain features of the orthogonal and unitary groups beyond β = 1 and β = 2 to general β > 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.