On idempotency of linear combinations of idempotent matrices

Abstract

P-1, P-2 and P-3 being any three different nonzero mutually commutative n x n idempotent matrices, and C-1 C-2 and C-3 being nonzero scalars, the problem of characterizing some situations, where a linear combination of the form P = c(1)P(1) + c(2)P(2) or P = c(1)P(1) + c(2)P(2) + c(3)P(3), is also an idempotent matrix has been considered. Moreover two interesting results about the idempotency of linear combinations of 2 x 2 idempotent matrices and 3 x 3 idempotent matrices have been given. A statistical interpretation of the idempotency problem considered in this study has also been pointed out. (C) 2003 Elsevier Inc. All rights reserved.