describe this constant in terms of the data matrix D?

Suppose that you are allowed to assume that at least one of the optimal solutions of the objective function in Exercise 3 must have mutually orthogonal columns in each of U and V , and in which each column of V is normalized to unit norm. (a) Use the optimality conditions of Exercise 3(a) to show that U must contain the largest eigenvectors of DDT in its columns and V must contain the largest eigenvectors of DT D in its columns. What is the value of the optimal objective function? (b) Show that the (length-normalized) optimal value for V that maximizes ||DV T ||2 F also contains the largest eigenvectors of DT D like (a) above. You are allowed to use the same assumption of orthonormal columns in V as above. What is the value of this optimal objective function? What does this tell you about the energy preserved by the SVD projection? (c) Show that the sum of the optimal objective function values in (a) and (b) is a constant that is independent of the rank k of the factorization but dependent only on D. How would you (most simply) describe this constant in terms of the data matrix D?

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