Complementary Projection is Projection

Theorem

Let $\HH$ be a Hilbert space.

Let $A$ be a projection.

Then the complementary projection $I - A$ is also a projection.

Proof

By Characterization of Projections, $A$ is Hermitian.

Then $\paren {I - A}^* = I^* - A^* = I - A$ from Adjoining is Linear.

So $I - A$ is also Hermitian.

From Complementary Idempotent is Idempotent, $I - A$ is idempotent.

Hence, applying Characterization of Projections, $I - A$ is a projection.

$\blacksquare$

Also

• Complementary Projection of Complementary Projection is Projection