Full answer: In Exercises 33-42 we use the notation that has been established in this

Chapter 7, Problem 40

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In Exercises 33-42 we use the notation that has been established in this section: A is an m X n matrix with positive singular values Ui. u2, , u rand zero singular values u r+l u r+Z , u n A has singular value decomposition A = UI V' where U is an m X m orthogonal matrix [u1 um], Vis an n X n orthogonal matrix [ v 1 v nl, and I is an m X n matrix with a diagonal upper left submatrix D of r positive entries that decrease in magnitude. The remaining entries of I are zeros. Properties of U, V, and I Prove that if A = DIV' is a singular value decomposition of A then ( a) the column vectors of V are eigenvectors of A'A ( b) the column vectors of U are eigenvectors of AA' ( c) the nonzero elements of I are the nonzero singular values of A.