Abstract: Let A be an m by n matrix of rank l, and let M and N be m by k and n by q matrices, respectively, where k is not necessarily equal to q or rank(M AN) < min(k, q). In this paper, we provide some necessary and sufficient conditions for the validity of the rank subtractivity formula: rank(A-AN(M AN)-M A) = rank(A)-rank(AN(M AN)-M A)by applying the full rank decomposition of A = F G(F ∈ Rm×l, G ∈ Rl×n, rank(A) =rank(F) = rank(G) = l) and the product singular value decomposition of the matrix pair[F M, GN ]. This rank subtractivity formula along with the condition under which it holds is called the extended Wedderburn-Guttman theorem. 

Key words: Rank equality, rank subtractivity, product singular value decomposition, generalized inverses