Rank formulae from the perspective of orthogonal projectors

Abstract

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.

Keywords:

• partitioned matrix
• dimension of subspaces
• Moore–Penrose inverse

AMS Subject Classifications::

• 15A03
• 15A57

Acknowledgements

The authors are grateful to an anonymous referee for his/her valuable suggestions on an earlier version of this article. Oskar Maria Baksalary would like to express his sincere thanks to the Alexander von Humboldt Foundation and the German Academic Exchange Service (DAAD) for their financial support.