Equalities for estimators of partial parameters under linear model with restrictions

Estimators of partial parameters in general linear models involve some complicated operations of the submatrices in the given matrices and their generalized inverses in the models. In this case, more efforts are needed to find variety of properties hidden behind these estimators. In this paper, we use some new analytical tools in matrix theory to investigate the connections between the ordinary least-squares estimators and the best linear unbiased estimators of the whole and partial unknown parameters in general linear model with restrictions. In particular, we derive necessary and sufficient conditions for the ordinary least-squares estimators to be the best linear unbiased estimators of the whole and partial unknown parameters in the model.