Idempotency of linear combinations of three idempotent matrices, two of which are commuting

Abstract

The considerations of the present paper were inspired by Baksalary [O.M. Baksalary, Idempotency of linear combinations of three idempotent matrices, two of which are disjoint, Linear Algebra Appl. 388 (2004) 67–78] who characterized all situations in which a linear combination P = c 1 P 1 + c 2 P 2 + c 3 P 3 , with c i , i = 1 , 2 , 3 , being nonzero complex scalars and P i , i = 1 , 2 , 3 , being nonzero complex idempotent matrices such that two of them, P 1 and P 2 say, are disjoint, i.e., satisfy condition P 1 P 2 = 0 = P 2 P 1 , is an idempotent matrix. In the present paper, by utilizing different formalism than the one used by Baksalary, the results given in the above mentioned paper are generalized by weakening the assumption expressing the disjointness of P 1 and P 2 to the commutativity condition P 1 P 2 = P 2 P 1 .