Complementary Idempotent is Idempotent
Jump to navigation
Jump to search
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.
$\blacksquare$
Also see
Sources
- 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next) $\text{II}.3.2 (a)$