The way I understand it is that: if U is orthogonal, i.e., its columns (or rows) form an orthonormal basis for the n-dimensional Euclidean (coordinate) space, then the matrix $UU^t$ is an orthogonal projection of vectors onto the column space of U.
But what if U were not orthogonal? Is $UU^t$ still a projection onto the column space of U, but that the projection need not be orthogonal? This would be my naive guess, so I'm hoping to clear up any misunderstandings I currently have.
Is the operator [I - $UU^t$] or maybe [$uu^t$-I] then an (orthogonal?) projection onto the space that is orthogonal to the column space of U?
Thanks,