A property of orthogonal projectors
J. K. Baksalary1 , O. M. Baksalary2 , T. Szulc3
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
3 Faculty of Mathematics and Computer Science, Adam Mickiewicz University, ul. Matejki 48/49, PL 60-769, Poznań , Poland
Linear Algebra and Its Applications, 354, 35-39 (2002)
It is shown that if P1 and P2 are orthogonal projectors, then a product having P1 and P2 as its factors is equal to another such product if and only if P1 and P2 commute, in which case all products involving P1 and P2 reduce to the orthogonal projector P1P2. This is a generalization of a result by Baksalary and Baksalary [Linear Algebra Appl. 341 (2002) 129], with the proof based on a simple property of powers of Hermitian nonnegative definite matrices.