Some properties of BLUE in a linear model and canonical correlations associated with linear transformations

Let (x, X[beta], V) be a linear model and let A' = (A'1, A'2) be a p - p nonsingular matrix such that A2X = 0, Rank A2 = p - Rank X. We represent the BLUE and its covariance matrix in alternative forms under the conditions that the number of unit canonical correlations between y1 ( = A1x) and y2 ( = A2x) is zero. For the second problem, let x' = (x'1, x'2) and let a g-inverse V- of V be written as (V-)' = (A'1, A'2). We investigate the reations (if any) between the nonzero canonical correlations {1 [greater, double equals] [varrho]1 [greater, double equals] ... [greater, double equals] [varrho]1 > 0} due to y1 ( = A1x) and y2 ( = A2x), and the nonzero canonical correlations {1 [greater, double equals] [lambda]1 [greater, double equals] ... [greater, double equals] [lambda]v+r > 0} due to x1 and x2. We answer some of the questions raised by Latour et al. (1987, in Proceedings, 2nd Int. Tampere Conf. Statist. (T. Pukkila and S. Puntanen, Eds.), Univ. of Tampere, Finland) in the case of the Moore-Penrose inverse V+ = (A'1, A'2) of V.