Projector

Projectors are Operator with a special property

Π^2=Π^\begin{gather*} \hat{\Pi}^2=\hat{\Pi} \end{gather*}

A projector operator (aka. a projector) is a projection that

Π^u=∣u⟩⟨u∣⟨u∣u⟩\begin{gather*} \hat{\Pi}_u=\frac{\left\lvert u \right\rangle\left\langle u \right\rvert}{\left\langle u|u \right\rangle} \end{gather*}

Usually, by convention, we pick the representative state ∣u⟩\left\lvert u \right\rangle to be normalized, i.e., ⟨u∣u⟩=1\left\langle u|u \right\rangle=1. So sometimes it can be written as

Π^u=∣u⟩⟨u∣if⟨u∣u⟩=1\hat{\Pi}_u=\left\lvert u \right\rangle\left\langle u \right\rvert\quad \text{if} \quad\left\langle u|u \right\rangle=1

once i apply it to a state vv, i get the projection of that state vv on state uu