15.1 Prove that for all rank-one matrices, . Hint: Use Theorem 15.5.
15.2 Suppose u1, u2, …, un and v1, v2, …, vn are orthonormal bases for Ρn. Construct the matrix A that transforms each vi into ui to give Av1 = u1, Av2 = u2, …, Avn = un.
15.3 Let . Determine value(s) for the aij so that A has distinct singular values.
15.4 Prove that .
15.5 Find the SVD of