Idempotent Operator

linear-algebra

Definition

Idempotent Operator

A idempotent operator is a linear operator $T:VoV$ such that

$$T^2=T.$$

Equivalently, applying the operator twice has the same effect as applying it once.

Relation to Projections

Every orthogonal projection is idempotent. More generally, any projection operator satisfies $T^2=T$.

If $T$ is idempotent, then its eigenvalues can only be $0$ or $1$.