On commutativity of projectors

The purpose of this paper is to revisit two problems discussed previously in the literature, both related to the commutativity property P1P2 = P2P1, where P1 and P2 denote projectors (i.e., idempotent matrices). The first problem was considered by Baksalary et al. [J.K. Baksalary, O.M. Baksalary, T. Szulc, A property of orthogonal projectors, Linear Algebra Appl. 354 (2002) 35–39], who have shown that if P1 and P2 are orthogonal projectors (i.e., Hermitian idempotent matrices), then in all nontrivial cases a product of any length having P1 and P2 as its factors occurring alternately is equal to another such product if and only if P1 and P2 commute. In the present paper a generalization of this result is proposed and validity of the equivalence between commutativity property and any equality involving two linear combinations of two any length products having orthogonal projectors P1 and P2 as their factors occurring alternately is investigated. The second problem discussed in this paper concerns specific generalized inverses of the sum P1 + P2 and the difference P1 − P2 of (not necessary orthogonal) commuting projectors P1 and P2. The results obtained supplement those provided in Section 4 of Baksalary and Baksalary [J.K. Baksalary, O.M. Baksalary, Commutativity of projectors, Linear Algebra Appl. 341 (2002) 129–142].