Banerjee  has recently given proofs of various properties of real symmetric idempotent matrices. The aim of this note is to give shorter proofs of some of these. Some of our proofs are suggested by the fact that idempotent matrices represent projection operators, and by the known properties of such operators (see e.g. Halmos ). The recent paper by Luther  overlaps this note in content; but has a different aim.
"On Idempotent Matrices." Ann. Math. Statist. 37 (1) 295 - 296, February, 1966. https://doi.org/10.1214/aoms/1177699624