Suppose a matrix nx d matrix A has an SVD decomposition that can be written as nxd 7 U_x 2 x V XXF (b) (20 points) rixd where the singular-values in E (resp. ) are greater than (resp. lesser than) some y E R. Show that (a) (20 points) nxn U__ x 2 x V nxr nxd UxU V = U x E x V x V = dxr 2. (40 points) Suppose a matrix nx d matrix A with rank r, and has an SVD rxr AUX Ex V rxd RXF nxd Let us suppose A gets "corrupted" by a n xd, noise-matrix E, and A = A + E. Suppose the corrupted-matrix A has an SVD XXF and 71X7 A = U x x VI nxd rxd Suppose , is obtained from E, by keeping only the top r-many entries (i.e. we zero-out all diagonal-values that not in the list of top r-many SVs). Let A = UXE X V. Show that ||A - AllF V8r x ||E||2.