Factorization of Separable and Patterned Covariance Matrices for Gibbs Sampling

Recently the Gibbs sample has become a very popular estimation technique especially in Bayesian Statistics. In order to implement the Gibbs sampler, matrix factorization must be computed which normally is not problematic. When the dimension of the matrices to be factored is large, computation time increases to an amount to merit special attention. I have found that when the matrices to be factored are separable or patterned, results from matrix theory can assist in computation time reduction