Rank formulae from the perspective of orthogonal projectors
Oskar Maria Baksalary, Götz Trenkler · Linear and Multilinear Algebra · 2011
For matrices F and G having the same number of rows and the orthogonal projectors P = FF † and Q = GG †, with F † and G † denoting the Moore–Penrose inverses of F and G, respectively, several formulae for ranks of various functions of F, G, P and Q are established. Besides a collection of original characterizations, many of which involve the ranks of F*G and (F : G) (which coincide with the ranks of PQ and P + Q, respectively), some properties known in the literature are reestablished in a generalized form. The variety of relationships considered shows that the approach utilized in the article, based on the partitioned representations of the projectors, provides a powerful tool of wide applicability.