Quantum Physics Division
Faculty of PhysicsAdam Mickiewicz University

When is a linear combination of two idempotent matrices the group involutory matrix?

J. K. Baksalary1 , O. M. Baksalary2

1 Faculty of Mathematics, Informatics and Econometrics, Zielona Góra University, ul. Podgórna 50, PL 65-246 Zielona Góra, Poland
2 Institute of Physics, Adam Mickiewicz University, ul. Umultowska 85, PL 61-614 Poznań, Poland

Linear and Miltlilinear Algebra, 54 429-435 (2006)
DOI: 10.1080/03081080500473028


Coll and Thome [Coll, C. and Thome, N., 2003, Oblique projectors and group involutory matrices. Applied Mathematics and Computation , 140 , 517-522] considered the problem of `when a linear combination c1P1 + c2P2 of nonzero different complex idempotent matrices P1 , P2 , with nonzero complex numbers c1 , c2 , is the group involutory matrix?' According to the solution provided therein as Theorem 1, it is possible in a finite number of cases, each characterized by definite values of scalars c1 and c2. In the present article, this problem is revisited and it is shown that the actual number of cases, in which a linear combination of interest is the group involutory matrix, is infinite and that there is certain freedom regarding values of c1 and c2.


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