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

Chapter 7, Problem 37

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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. SVD and Orthogonal Matrices If A is an m X n matrix and P and Qare orthogonal matrices such that PAQ exists prove thatA and PAQ have the same singular values.