We present two proofs. From Marsaglia and Styan (1974, Theorem 5, p. 274) we have

for any generalized inverse A⁻. Because M is symmetric idempotent and V is nonnegative definite we have rank(MVM) = rank(MV). We are, therefore, asked to show that rank(MV) + rank(A) = rank(V) iff rank(A : V) = rank(V). This follows at once from (1) replacing A⁻ with A⁺. When W is nonnegative definite (A′WA)⁺A′W is a generalized inverse of A, and so rank(MVM) + rank(A) = rank(V) [hArr ] col(A) ⊆ col(V) with M := I_n − A(A′WA)⁺A′W. This completes our first proof. We see that W need only be nonnegative definite, not necessarily positive definite.

For our second proof we use (2.6) in Werner and Yapar (1996) (see also Werner, 1985, Lemma 3.2):

where [oplus ] indicates direct sum and where Q is any matrix such that col(Q) = null(A′), with null(·) denoting the null space of (·). In view of (2), clearly

iff

Because col(M) = null(A′), M = M′, V ≥ 0, MVM ≥ 0, we have rank(MVM) = rank(VM) = rank(VQ). As (3) is equivalent to rank(A) + rank(VQ) = rank(V), the first assertion is established. To see that an equivalent formulation of the problem is obtained when M := I_n − A(A′WA)⁺A′W, with W nonnegative definite and MVM changed to MVM′, we note that MVM′ ≥ 0, rank(MVM′) = rank(VM′), and col(M′) = null(A′) (see, e.g., Werner and Yapar, 1996, Theorem 2.5 and (2.2) and (2.3)). Therefore, col(VM′) = col(VQ), and the assertion again is an immediate consequence of (3) [hArr ] (4).

It is easy to see that the preceding equivalent formulation can be generalized further to any square (not necessarily nonnegative definite) matrix W such that rank(A′WA) = rank(A) for then (A′WA)⁻A′W is a generalized inverse of A and col(M′) = null(A′).