On the Wedderburn–Guttman theorem

Let A be a u by v matrix of rank a, and let M and N be u by g and v by g matrices, respectively, such that M′AN is nonsingular. Then, rank(A − N(M′AN)−1M′A) = a − g, where g = rank(AN(M′AN)−1M′A) = rank(M′AN). This is called Wedderburn–Guttman theorem. What happens if M′AN is rectangular and/or singular? In this paper we investigate conditions under which the regular inverse (M′AN)−1 can be replaced by a g-inverse (M′AN)− of some kind, thereby extending the Wedderburn–Guttman theorem. The resultant conditions look similar to those arising in seemingly unrelated contexts, namely Cochran’s and related theorems on distributions of quadratic forms involving a normal random vector.