Given the matrix $$\begin{bmatrix} 3 & 3 & 0 & 0 \\ 8/3& 2 & 2 & 0 \\ 0 & 3/4 & 1 & 3/4 \\ 0 & 0 & 3/2 & 1 \end{bmatrix}$$
I want to calculate the biggest eigenvalue using the power iteration method. Since it is an iterative method, I need an initial vector to start, how can I choose that vector so that it converges to the eigen value for sure?
Is is only a problem in exact arithmetic. When using precision arithmetic, rounding errors will inevitably infuse the current vector with a component in the direction of the dominant eigenvector. Subsequent iterations will magnify the relative significance of this component until it is dominant and the new for all intent and purposes parallel with the eigenvector corresponding to the dominant eigenvalue. You can typically use a vector of ones or a vector consisting of random numbers to initialize the search.