Rank formulae from the perspective of orthogonal projectors
O. M. Baksalary1 , G. Trenkler2
1 Faculty of Physics, Adam Mickiewicz University, ul. Umultowska 85, PL 61-614 Poznań, Poland
2 Department of Statistics, Dortmund University of Technology, Vogelpothsweg 87, D-44221 Dortmund, Germany
Linear and Multilinear Algebra, 59, 607-625 (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.