# Complementary Idempotent is Idempotent

## Theorem

Let $\HH$ be a Hilbert space.

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

Let $A$ be an idempotent operator.

Then the complementary idempotent $I - A$ is also idempotent.

## Proof

 $\ds \paren {I - A}^2$ $=$ $\ds I^2 - I A - A I + A^2$ $\ds$ $=$ $\ds I^2 - 2 A + A^2$ Definition of Identity Operator $\ds$ $=$ $\ds I - A$ Definition of Idempotent Operator

That is, $I - A$ is idempotent.

$\blacksquare$