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(a) Implement power iteration to compute the dominant eigenvalue and a corresponding normalized eigenvector of the matrix. $A=⎣⎡ 2103 336 241 ⎦⎤ $ As starting vector, take $x_{0}=[0 0 1 ]_{T}$. (b) Using any of the methods for deflation given in section 4.5.4, deflate out the eigenvalue found in part $(a)$ 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 $a$ and $b$.

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 np

def power_iteration(A, x0, max_iterations):

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