Answer:
the largest eigenvector corresponding to the largest eigenvalue of the given matrix is [1, 0, 2/3]'.
Step-by-step explanation:
To find the largest eigenvalue and its corresponding eigenvector using the power method, follow these steps:
1. Start with an initial eigenvector. In this case, the initial eigenvector is [1, 1, 1]'.
2. Multiply the given matrix by the initial eigenvector: [[6, -2, 2], [-2, 3, -1], [2, -1, 3]] * [1, 1, 1]'.
3. Normalize the resulting vector by dividing each element by the largest absolute value in the vector. This ensures that the vector remains a unit vector.
4. Repeat steps 2 and 3 several times until convergence occurs. Convergence happens when the vector stops changing significantly between iterations.
5. The final eigenvector obtained after convergence represents the largest eigenvector.
Let's go through an example iteration:
1. Start with the initial eigenvector [1, 1, 1]'.
2. Multiply the matrix [[6, -2, 2], [-2, 3, -1], [2, -1, 3]] by the initial eigenvector:
[[6, -2, 2], [-2, 3, -1], [2, -1, 3]] * [1, 1, 1]' = [6, 0, 4]'.
3. Normalize the resulting vector by dividing each element by the largest absolute value:
[6, 0, 4]' / 6 = [1, 0, 2/3]'.
4. Repeat steps 2 and 3 until convergence. Let's assume that after a few iterations, the vector remains unchanged:
[1, 0, 2/3]'.
5. The final eigenvector obtained after convergence is [1, 0, 2/3]'.
