I was trying to think of the reason behind the PCA eigen vectors being transposed, that is, why $y= U_D^T x$.
Geometrically, is it because since we are taking the projection of the point $x$ into a reduced dimension hyperplane, we are essentailly saying that the component of x which is already in the required hyperplane need not be changed in any way, but the ones which are lying outside the hyperplane at a certain angle will need to have their dot products taken as part of their being projected on to the plane?
What would be the interpretation in terms of maximizing variance?