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
, parametrized by β > 0, a > -1,and b > -1, is the probability distribution whose density is proportional to
. 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,
, 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. 相似文献
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