Complementary Projection is Complementary Idempotent
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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.
$\blacksquare$