Alternative characterizations of the extended Wedderburn-Guttman theorem

Let A be a u by v matrix of rank a, and let M and N be u by p and v by q matrices, respectively, where p is not necessarily equal to q or rank(M'AN) < min(p, q). Takane and Yanai [Y. Takane, H. Yanai, On the Wedderburn-Guttman theorem, Linear Algebra Appl. 410 (2005) 267-278] investigated the conditions under which rank(A - AN(MAN)-M'A) = rank(A) - rank(AN(M'AN)-M'A). This is called the extended Wedderburn-Guttman theorem. In this paper, we give alternative characterizations of these conditions using the product singular value decomposition (PSVD) of matrix triplets. (c) 2006 Elsevier Inc. All rights reserved.