# Complementary Projection is Complementary Idempotent

## Theorem

Let $\HH$ be a Hilbert space.

Let $A$ be a projection.

Let $B$ be the complementary projection of A.

Then $B$ is the complementary idempotent of $A$.

## Proof

By the definition of projection, $A$ is an idempotent operator.

The result follows immediately from the definitions of:

where the constructions of the complementary projection and the complementary idempotent from $A$ are identical.

$\blacksquare$