I'm trying to understand dominant eigenvalues and I found this website that has a explanation of it (Definition 9.2). In the example, the power method is used to find the dominant eigenvector which correspondes to the eigenvalue of 1.
When I calculate the eigenvalues and vectors of the matrix in the example, I got this result:
The first line are the eigenvalues and the second row the eigenvectors. As you can see, the eigenvector that corresponde to to the eigenvalue of 1 is
{-0.577, -0.577, -0.577}
If I calculate the powers of the matrix, I find that after M^9, it converges as shown in the website
I don't understand what is the difference between the eigenvector that I found that corresponde to to the eigenvalue of 1 and the eigenvector that is found after elevating the matrix many times, and that the website described also as the eigenvector of eigenvalue 1.