Convergence of the Inverse Iteration on Symmetric Matrices.

Algorithm 6.32: Inverse Iteration

Consider the inverse iteration (Algorithm 6.32) applied to a real symmetric matrix A Rmxn. Suppose is the closest eigenvalue to and L s the second closest, that is. Let q1,... ,7n denote eigenvectors corresponding the eigenvalues of A. Suppose further we have an initial vector T such that K0. Prove that, at iteration k, the vector bk) in the inverse iteration converges as k)- You may consider taking the following steps (i) Suppose all the eigenvectors qi, q2,..., 7n are orthonormal, write down the eigen- decomposition of (A -I)1 (ii) Use induction to show that the inverse iteration method produces a sequence of vectors bk), k- 1,2,...in the form of (A-111)-kb(0) jk) _ (iii) Use the results in (i) and (ii) to complete the rest of the proof Input: Matrix A E Rnxn, an initial vector b(0) Rn where lla and a shift scalar E R Output: An eigenvalue (m) and its eigenvector b(m 1, 1: for k- 1,2,... , m do 2: 3 4: 5: end for Solve (A-1h(k)-jki) for u,(k) Apply (A-11)-1 DNormalise DEstimate eigenvalue