$$A = \begin{pmatrix}4 &–1 & 0 \\-1& 4& -1\\ 0 &-1& 4 \end{pmatrix} $$ $X_0 = (1, 1, 1)$. Perform 5 steps of power iterations to find the largest eigenvalue of A.
And my solution is this.
But, some people says that eigenvalue is 5. This is because when last step, it is based on the middle value. Why?
You can check with any entry you want, it is indeed approximately $5$ (if you work it out analytically you notice that largest eigenvalue is $4+ \sqrt{2}$), but remember that you are using an approximation method, so there is no "correct"way to do it. If you would use more iterations we expect the error to get smaller and to approach the "true" eigenvalue, if it exists of course.
Another, better way to do it is by looking at: http://college.cengage.com/mathematics/larson/elementary_linear/5e/students/ch08-10/chap_10_3.pdf
Where they show that a good way to do it is by computing the so-called Rayleigh quotient, which if $A^5 x= \bar{x} $ is defined to be:
$$ \frac{A\bar{x} \cdot\bar{x}}{x \cdot x} $$
Give it a try and let us know which $\lambda$ you find ?