Idempotency of linear combinations of two idempotent matrices Original Research Article

A complete solution is established to the problem of characterizing all situations, where a linear combination of two different idempotent matrices P1 and P2 is also an idempotent matrix. Including naturally three such situations known in the literature, viz., if the combination is either the sum P1+P2 or one of the differences P1−P2, P2−P1 (and appropriate additional conditions are fulfilled), the solution asserts that in the particular case where P1 and P2 are complex matrices such that P1−P2 is Hermitian, these three situations exhaust the list of all possibilities and that this list extends when the above assumption on P1 and P2 is violated. A statistical interpretation of the idempotency problem considered in this note is also pointed out.