2. [(ii): The V's are eliminated through G,. Since the r + m row vectors of the matrices Y and Z may be assumed to be linearly independent, any such set of vectors can be transformed into any other through an element of G3 .] 2. (i) If p < r + m, and V = Y'Y, S = Z'Z, the p X P matrix V + S is nonsingular with probability 1, and the characteristic roots of the equation (61) IV - X( V + S) I = 0 constitute a maximal set of invariants under G1, G2 , and G3 . (ii) Of the roots of (61), p - min(r, p) are zero and p - min(m, p) are equal to one. There are no other constant roots, so that the number of variable roots, which constitute a maximal invariant set, is min (r, p) + min(m, p) - p . [The multiplicity of the root X= 1 is p minus the rank of S, and hence p - min(m, p). Equation (61) cannot hold for any constant A'" 0,1 for almost all V, S, since for any p. '" 0, V + p.S is nonsingular with probability 1.]