[[Linear endomorphism]]
# Projection operator

A **projection operator** $P : V \to V$ is an [[Idempotent]] [[linear endomorphism]],
i.e. an operator whose restriction $P \restriction PV$ to its image is the identity. #m/def/linalg 
$$
\begin{align*}
P^2v = Pv
\end{align*}
$$
The geometric intuition is that a projection operator _projects_ vectors into the subspace $PV$.

## Properties

- In [[bra-ket notation]] the projection operator onto the space spanned by normalized $\ket{v}$ is $\ket{v}\bra{v}$.

#
---
#state/tidy | #lang/en | #SemBr