Matrix spaces and ordinary least square estimators in linear models for random matrices

Abstract

This article, using generalized inverses of matrices, studies the spaces whose elements are matrices. Based on the results obtained, for linear models for random matrices, the article explores the role of ordinary least square estimators in identifying linear estimable functions, and the role of minimum norm ordinary least square estimators in creating linear unbiased estimators. With added conditions on the covariance matrix for vectorized response, it is shown that a linear unbiased estimator constructed from the minimum norm ordinary least square estimator is a best linear unbiased estimator with respect to the risk induced from squared distance loss.