So I'm doing Inverse Iteration in Excel to find the dominant eigenvalue and eigevector of a matrix.
This particular method involves estimating an eigenvalue, multiplying the identity matrix by it, then subtracting from the original matrix.
From this point, the iteration begins by multiplying the inverse matrix with an intial vector
$v_0 = $
$ \begin{matrix} 1 \\ 1 \\ 1 \\ \end{matrix} $
to find $u_1$
Now the next step is normally ok, but in this case it confused me
Normally you would normalise $u_1$ by dividing by the maximum value in the vector, for example -
$u_1 = $
$ \begin{matrix} 1 \\ 2 \\ 3 \\ \end{matrix} $
Then you would normalise by dividing the entire vector by 3
However in this case
$u_1 =$
$ \begin{matrix} 0.181818 \\ -0.66667 \\ 0.109091 \\ \end{matrix} $
The method says that the maximum value of this vector is the -0.66667, why is this the case? Is it always the case that you look at the number regardless of the sign to determine the maximum value?
Let me know if you need any clarifications.