Projection Matrix

The matrix $P$ is a projection matrix if $PP=P$. Why would this mean the matrix is a projection? Well if $P\pmb x = \pmb y$ and $P$ is a projection matrix then $PP\pmb x = P\pmb x = \pmb y$. Or written another way $PP\pmb x= P\pmb y = \pmb y$. So a projection matrix moves a vector to a location and after it’s been moved to that location, moving it to that location again doesn’t do anything. Hence if a linear transformation defined by $P$ is idempotent (i.e. $PP = P$) then it is a projection matrix.