In this paper we consider the estimation of the expectation vector
Xβ under the general linear model
{y,Xβ,σ2V}. We introduce a
new handy representation for the rank of the difference of the covariance
matrices of the ordinary least squares estimator OLSE(Xβ) (=
Hy, say) and the best linear unbiased estimator
BLUE(Xβ) (= Gy, say). From this formula some
well-known conditions for the equality between Hy and Gy
follow at once. We recall that the equality between Hy and
Gy can be characterized by the rank-subtractivity ordering
between the covariance matrices of y and Hy. This rank
characterization suggests a particular presentation for the rank of the
difference of the covariance matrices of Hy and Gy. We
show, however, that this presentation is valid if and only if the model is
connected.