Annihilation Technique Suppose the n n matrix A has eigenvalues 1,... ,n ordered by
Chapter 9, Problem 9.2.18(choose chapter or problem)
Annihilation Technique Suppose the n n matrix A has eigenvalues 1,... ,n ordered by |1| > |2| > |3||n |, with linearly independent eigenvectors v(1) , v(2) ,... , v(n) . a. Show that if the Power method is applied with an initial vector x(0) given by x(0) = 2v(2) + 3v(3) ++ n v(n) , then the sequence {(m) } described in Algorithm 9.1 will converge to 2. b. Show that for any vector x = n i=1 iv(i) , the vector x(0) = (A 1 I)x satisfies the property given in part (a). c. Obtain an approximation to 2 for the matrices in Exercise 1. d. Show that this method can be continued to find 3 using x(0) = (A 2 I)(A 1 I)x
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