Consider the linear model Yij = P, + Qi + 'f/j + eij, i =

1, ... ,a, j = 1, ... ,b. As in Section 1, we can write the design matrix as X =

[Xo, Xl"'" X

a, Xa+l, ... , Xa+b]' If we write the observations in the usual order, we can use Kronecker products to write the design matrix. Write X = [J, X*, X**], where X* = [Xl, ... , Xa], and X** = [Xa+l,"" Xa+b]'

Using Kronecker products, X* = [Ia0Jb], and X** = [Ja0h]. In fact, with n = ab, J = I n = [Ja 0 Jb]. Use Kronecker products to show that X: (I -

[l/n]J~)X** = O. In terms of Section 1, this is the same as showing that C(ZI,"" Za) .1 C(Za+l,"" Za+b)' Also show that [(l/a)J~ 0 h] is the perpendicular projection operator onto C(X**) and that M'I = [(l/a)J~ 0

(h - (l/b)Jg)].