4 (Matrix Norms and Approximations, 7.5 points each). (a) Show that the Schatten p-norm is unitarily invariant for any p > 0 even though it is only truly a norm for p 1. (b) (5 points) Let A Rmn (n > m) with compact SVD A = m i=1 iuiv> i . Consider the rank-r (m > r) approximation problem: min X X AF , s.t. rank(X) r. Show that one optimal solution is the truncated SVD of the following form Ar = r i=1 iuiv> i . Moreover, show that the solution is unique whenever r > r+1 (a brief justification is sufficient)