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(Solved): (a) Implement power iteration to compute the dominant eigenvalue and a corresponding normalized eig ...
(a) Implement power iteration to compute the dominant eigenvalue and a corresponding normalized eigenvector of the matrix. As starting vector, take . (b) Using any of the methods for deflation given in section 4.5.4, deflate out the eigenvalue found in part and apply power iteration again to compute the second largest eigenvalue of the same matrix. (c) Use a general real eigensystem library routine to compute all of the eigenvalues and eigenvectors of the matrix, and compare the results with those obtained in parts and .
Expert Answer
Part a) ::To implement power iteration for computing the dominant eigenvalue and a corresponding normalized eigenvector of a matrix, we can follow these steps:Initialize the starting vector x_0.Set the number of iterations, max_iterations.For each iteration from 1 to max_iterations, do the following:Compute the matrix-vector product y = A * x_{k-1}.Normalize the vector y by dividing each element by the maximum absolute value in y.Update the eigenvector estimate x_k = y.Compute the eigenvalue estimate by taking the dot product of A * x_k and x_k.4. Return the dominant eigenvalue and the corresponding normalized eigenvector.Let's apply these steps to the given matrix A:import numpy as npdef power_iteration(A, x0, max_iterations):