Projection definition in Linear Algebra Why is a mathematical projection defined as $P^2 = P$? I understand that it might be because once a vector has been projected onto a space, if projected again, it should give the same thing. Is there anything more to it?

If,  $ E^n = V \oplus W $, then,
$Px=x \;\;\forall x\in V $
$Px=0 \;\;\forall x \in W$
If dimensionality of $V$ is $r$, you will find that, there exists matrices such that,
$P=T\Delta_rT^{-1}$ where $\Delta_r$ is a diagonal matrix with first $r$ diagonals equal to $1$ and rest other are $0$.
So what we get,
$P^2=(T\Delta_rT^{-1})(T\Delta_rT^{-1})=T\Delta_rT^{-1}=P$
