Idempotence and Orthogonality in Relation to Mixed Model Adjustments

Abstract

The purpose of this paper is to look into several common idempotent matrices of interest in the context of any mixed model least-squares adjustment. The cofactor (scaled covariance) matrices, based on the law of error propagation, are incorporated into the least-squares estimation solutions. The rank of an idempotent matrix is given by taking the matrix trace, which leads to an unbiased estimate of the unit weight variance. The orthogonality between two projected vector subspaces is proven by using the generalized inner products, or equivalently, according to the cosine law. The Lagrange multiplier vector and its associated cofactor matrix are then employed to verify the already determined idempotent as well as orthogonal properties, reflecting the correctness of the formulated matrix equations.