Projection Transformation

#Math

$\displaystyle \mathrm{Proj}(y;A)=\text{argmin}{v\in \mathcal{R}(A)}\lVert v-y\rVert{2}=\mathbf{A}(\mathbf{A}^{\top}\mathbf{A})^{-1}\mathbf{A}^{\top}y$

• Projection of a vector $\displaystyle y$ onto the a matrix $\displaystyle A$'s columnspace

$\displaystyle \text{Proj}(y;a)=\frac{aa^{\top}}{a^{\top}a}y$

• Projection of a vector $\displaystyle y$ onto another vector $\displaystyle a$