Projection

'''The page below may have some issues (i.e., the norm statement). See Projector for a possibly better page'''

If $$A$$ is skinny and full rank, then the projection operation is $$\mathcal P_{\mathcal R(A)}(y) = A(A^TA)^{-1}A^T y$$

Properties of projection matrix $$P$$
TBC
 * If $$P$$ is a projection matrix, then so is $$I-P$$.
 * The norm of a projection matrix is either 1 or 0 either 1 or 0
 * Assume that $$\mathbf{P}$$ is a matrix that projects a given vector to the column space of a matrix $$\mathbf{A}$$. The projected version $$\mathbf{Pv}$$ of a vector $$\mathbf v$$ satisfies $$||\mathbf{Pv-v}||=||\mathbf{v}||$$ iff $$\mathbf{v}\in \mathcal{N}(\mathbf A^T)$$ (i.e., $$v$$ must be orthogonal to every column of $$\mathbf A^T$$ -- easy to verify by drawing an example vector).