Recall that eigenvectors, once normalized, are only unique up to a sign, that is, if Ax = sx, then A(−x) = s(−x). In the SVD where A = UDVT , if we insist that the diagonal elements of D are...

Recall that eigenvectors, once normalized, are only unique up to a sign, that is, if Ax = sx, then A(−x) = s(−x). In the SVD where A = UDVT , if we insist that the diagonal elements of D are nonnegative, can we still choose some arbitrary signs for the left and right vectors? For the complex SVD, show that the singular vectors can be scaled using an imaginary number while still ensuring that the diagonal elements of D are nonnegative. Suppose we computed two equivalent complex decompositions, A = U1DVH 1 = U2DVH 2 ; what are the relationships among U1, U2, V1, and V2?