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

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 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.