On idempotency of linear combinations of idempotent matrices

摘要

P1, P2 and P3 being any three different nonzero mutually commutative n×n idempotent matrices, and c1,c2 and c3 being nonzero scalars, the problem of characterizing some situations, where a linear combination of the form P=c1P1+c2P2 or P=c1P1+c2P2+c3P3, is also an idempotent matrix has been considered. Moreover two interesting results about the idempotency of linear combinations of 2 × 2 idempotent matrices and 3 × 3 idempotent matrices have been given. A statistical interpretation of the idempotency problem considered in this study has also been pointed out.