Quantum Physics Division

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Commutativity of projectors

J. K. Baksalary^{1}
,
O. M. Baksalary^{2}

^{1} Faculty of Mathematics, Informatics, and Econometrics, Zielona Góra University, ul. Podgórna 50, PL 65-246 Zielona Góra, Poland

^{2} Department of Physics, A. Mickiewicz University, 61-614 Poznań, Poland

**Linear Algebra and Its Applications, 341, 129-142 (2002)**

**DOI:** 10.1016/j.laa.2006.02.019

Abstract:

The purpose of this paper is to revisit two problems discussed previously in the literature, both related to the commutativity property P_{1}P_{2} = P_{2}P_{1}, where P_{1} and P_{2} 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 P_{1} and P_{2} are orthogonal projectors (i.e., Hermitian idempotent matrices), then in all nontrivial cases a product of any length having P_{1} and P_{2} as its factors occurring alternately is equal to another such product if and only if P_{1} and P_{2} 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 P_{1} and P_{2} as their factors occurring alternately is investigated. The second problem discussed in this paper concerns specific generalized inverses of the sum P_{1} + P_{2} and the difference P_{1} − P_{2} of (not necessary orthogonal) commuting projectors P_{1} and P_{2}. 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].

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