Least-Squares Solution (LSS) of a linear matrix equation and Ordinary Least-Squares Estimator (OLSE) of unknown parameters in a general linear model are two standard algebraical methods in computational mathematics and regression analysis. Assume that a symmetric quadratic matrix-valued function Φ(Z) = Q − ZPZ0 is given, where Z is taken as the LSS of the linear matrix equation AZ = B. In this paper, we establish a group of formulas for calculating maximum and minimum ranks and inertias of Φ(Z) subject to the LSS of AZ = B, and derive many quadratic matrix equalities and inequalities for LSSs from the rank and inertia formulas. This work is motivated by some inference problems on OLSEs under general linear models, while the results obtained can be applied to characterize many algebraical and statistical properties of the OLSEs.
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A general linear model can be given in certain multiple partitioned forms, and there exist submodels associated with the given full model. In this situation, we can make statistical inferences from the full model and submodels, respectively. It has been realized that there do exist links between inference results obtained from the full model and its submodels, and thus it would be of interest to establish certain links among estimators of parameter spaces under these models. In this approach the methodology of additive matrix decompositions plays an important role to obtain satisfactory conclusions. In this paper, we consider the problem of establishing additive decompositions of estimators in the context of a general linear model with partial parameter restrictions. We will demonstrate how to decompose best linear unbiased estimators (BLUEs) under the constrained general linear model (CGLM) as the sums of estimators under submodels with parameter restrictions by using a variety of effective tools in matrix analysis. The derivation of our main results is based on heavy algebraic operations of the given matrices and their generalized inverses in the CGLM, while the whole contributions illustrate various skillful uses of state-of-the-art matrix analysis techniques in the statistical inference of linear regression models.
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