Chapter 7.4, Problem 19E

In Exercises 17–24, A is an m × n matrix with a singular value decomposition A = UΣV^T, where U is an m × m orthogonal matrix, Σ is an m × n “diagonal” matrix with r positive entries and no negative entries, and V is an n × n orthogonal matrix. Justify each answer.

19. Show that the columns of V are eigenvectors of A^T A, the columns of U are eigenvectors of AA^T, and the diagonal entries of Σ are the singular values of A. [Hint: Use the SVD to compute A^TA and AA^T.]