Projection/Idempotent Linear Operators

Projection/Idempotent Linear Operators

Definition: Let $X$ be a linear space. A linear operator $P : X \to X$ is said to be a Projection or Idempotent if $P^2 = P$, that is, $P(P(x)) = P(x)$ for every $x \in X$.

If $X$ is a linear space and $P : X \to X$ is a projection, then the kernel of $P$ has a nice form which is proven in the lemma below.

Lemma 1: Let $X$ be a linear space and let $P : X \to X$ be a projection. Then $(I - P)(X) = \ker P$.
  • Proof: Let $x \in (I - P)(X)$. Then there exists an $x' \in X$ such that $x = (I - P)(x') = I(x') - P(x') = x' - P(x')$. Since $P^2 = P$, we have that:
(1)
\begin{align} \quad P(x) = P(x' - P(x')) = P(x') - P^2(x') = P(x') - P(x') = 0 \end{align}
  • Therefore $x \in \ker P$. So:
(2)
\begin{align} \quad (I - P)(X) \subseteq \ker P \end{align}
  • The reverse inclusion is trivial:
(3)
\begin{align} \quad (I - P)(X) \supseteq \ker P \end{align}
  • Therefore:
(4)
\begin{align} \quad (I - P)(X) = \ker P \quad \blacksquare \end{align}
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