4.6 (The Power Method). In this exercise, we derive how to compute eigenvec- tors (and hence singular vectors) using the power method. Let I' 6 Rnx\" be a symmetric positive semidenite matrix. Let qo be a random vector that is uni- formly distributed on the sphere S\"_1 (we can generate such a random vector by taking an n-dimensional iid N0), 1) vector and then normalizing it to have unit 2 norm). Generate a sequence of vectors q1,q2, . . . via the iteration PQk q = _ 4.7.9 \"1 urging ( ) This iteration is called the power method. Suppose that there is a gap between the rst and second eigenvalues of I': A10") > A2(I'). 1 What does qk converge to? Hint: write I' = VAV\" in terms of its eigenvec tors/values. How does V*qk evolve? A 2 Obtain a bound on the error \"an.c qoollg in terms of the spectral gap 5%? 3 Your bound in 2 should suggest that as long as there is a gap between A1 and A2, the power method converges rapidly. How does the method behave if A1 = A2 1? 4 How can we use the power method to compute the singular values of a matrix X 6 IR\""'\"\"'2