idempotent

An element e of any algebra is called idempotent if e^2 = e — squaring it does nothing new. The word means roughly same power: applying it again gives the same thing. In the algebra End(V) of operators, the idempotents are exactly the projection operators, so idempotent is the abstract, algebra-level name for what a projection is concretely.

The algebraic constraint forces the spectrum. If e^2 = e then e satisfies x^2 - x = 0, i.e. x(x - 1) = 0, so every eigenvalue must be a root of that polynomial: the only possible eigenvalues are 0 and 1. Eigenvectors with eigenvalue 1 are the vectors e fixes (the image), and eigenvectors with eigenvalue 0 are the vectors e annihilates (the kernel). Since 0 and 1 are distinct simple roots, a nontrivial idempotent operator is automatically diagonalizable, with diagonal made of 0s and 1s.

This is why projections are so well behaved: their entire structure is forced by one equation. The rank of e equals the number of 1s, which equals the dimension of the subspace it projects onto, and the trace of e equals that same rank. There is no Jordan complication, no defective eigenvalues — idempotency rules them out.

Idempotents organize an algebra. A pair e and 1 - e are both idempotent, multiply to zero, and sum to the identity; this is the algebraic fingerprint of splitting the underlying space into a direct sum. Families of mutually orthogonal idempotents summing to 1 decompose the space into several pieces at once, which is exactly how the spectral projections of a diagonalizable operator work.