Complementary Idempotent is Idempotent

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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.


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

That is, $I - A$ is idempotent.


Also see