Homework 4 1. Let H = I − 2uu T (a) Show that HH T = I (b) Show that if B = HAH T then A and B have the same eigenvalues 2. Let A =   10 2 1 2 3 1 1 1 −3   (a) Find the first three iterations of the power method applied to A. (b) Find the first three iterations of the symmetric power method applied to A (c) Suppose you wish to apply the inverse power iteration to find the two other eigenvalues of A. i. What is the α value that you would choose for each case? Justify your choice. ii. Apply two iterations of the inverse power method using one of your α choices from the previous part. What is the approximation to the eigenvalue of A that you get ? (d) Apply Householder Transformation to reduce A to a tridiagonal matrix. 3. Let G θ = ? cos(θ) sin(θ) −sin(θ) cos(θ) ? (a) If G θ x = y, show that x and y have the same length. (b) If x = [x 1 ,x 2 ] T and y = [y 1 ,0], find G θ . 1