# Complementary Projection of Complementary Projection is Projection

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## Theorem

Let $\HH$ be a Hilbert space.

Let $I$ be an identity operator on $\HH$.

Let $A$ be a projection.

Let $B$ be the complementary projection of A.

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

## Proof

From Complementary Projection is Projection the complementary projection of $B$ is well-defined.

Let $C$ be the complementary projection of $B$.

We have:

\(\ds C\) | \(=\) | \(\ds I - B\) | Definition of Complementary Projection | |||||||||||

\(\ds \) | \(=\) | \(\ds I - \paren{I - A}\) | Definition of Complementary Projection | |||||||||||

\(\ds \) | \(=\) | \(\ds A\) |

$\blacksquare$