Some matrix results related to a partitioned singular linear model

Consider the linear model {y, X beta, V}, where the model matrix X may not have a full column rank and V might be singular. In this paper we introduce a formula for the difference between the BLUEs of X beta under the full model and the model where one observation has been deleted. We also consider the partitioned linear regression model where the model matrix is (X(1) : X(2)) the corresponding vector of unknown parameters being (beta(1)' : beta(2)')'. We show that the BLUE of X(1) beta(1) under a specific reduced model equals the corresponding BLUE under the original full model and consider some interesting consequences of this result.