Linear Algebra/Topic: The Method of Powers/Solutions

Solutions Problem 1 Use ten iterations to estimate the largest eigenvalue of these matrices, starting from the vector with components and . Compare the answer with the one obtained by solving the characteristic equation. <li> <li> Answer <li> The largest eigenvalue is . <li> The largest eigenvalue is . Problem 2 Redo the prior exercise by iterating until has absolute value less than At each step, normalize by dividing each vector by its length. How many iterations are required? Are the answers significantly different? Problem 3 Use ten iterations to estimate the largest eigenvalue of these matrices, starting from the vector with components , , and . Compare the answer with the one obtained by solving the characteristic equation. <li> <li> Answer <li> The largest eigenvalue is . <li> The largest eigenvalue is . Problem 4 Redo the prior exercise by iterating until has absolute value less than . At each step, normalize by dividing each vector by its length. How many iterations does it take? Are the answers significantly different? Problem 5 What happens if ? That is, what happens if the initial vector does not to have any component in the direction of the relevant eigenvector? Answer In theory, this method would produce . In practice, however, rounding errors in the computation introduce components in the direction of , and so the method will still produce , although it may take somewhat longer than it would have taken with a more fortunate choice of initial vector. Problem 6 How can the method of powers be adopted to find the smallest eigenvalue? Answer Instead of using , use .