Definition:Complementary Idempotent

Definition
Let $H$ be a Hilbert space.

Let $A \in B \left({H}\right)$ be an idempotent operator.

Then the complementary idempotent (operator) of $A$ is the bounded linear operator $I - A$, where $I$ is the identity operator on $H$.

The name is appropriate, by Complementary Idempotent is Idempotent and the fact that $I - \left({I - A}\right) = A$.

Complementary Projection
If $A$ is a projection, $I - A$ is called the complementary projection.

This name is justified by Complementary Projection is Projection.

Also see

 * Complementary Idempotent is Idempotent
 * Complementary Projection is Projection
 * Range of Idempotent is Kernel of Complementary Idempotent