Quantum Physics Division

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

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} 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

Abstract:

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 c_{1}P_{1} + c_{2}P_{2} of nonzero different complex idempotent matrices P_{1} , P_{2} , with nonzero complex numbers c_{1} , c_{2} , 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 c_{1} and c_{2}. 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 c_{1} and c_{2}.

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