I'm slightly confused about how to use the power method and the steps to calculate an eigenvalue. - I understand that the power method is defined as U(x+1) = AU(x)/a(x) where "a" is the first component of U(x). I do not understand at all what "U" is. Are we picking any vector we want that minimizes the error? What would I do given the practice problem below?
Apply the power method to $$ \begin{matrix} 1 & 2 \\ 3 & -4 \\ \end{matrix} $$
to obtain three approximations of the largest eigenvalue of A. What is the limiting vector u∞?
The key is, that $U$ - whatever initial value you take - converges to the first eigenvector (the eigenvector to the largest eigenvalue). So you simply start with $U=[1,0,0,...0]$ and iterate.
There is one caveat: the largest eigenvalue must (in absolute value) be greater than 1 and distinct from the second highest.
Here is a table of iterations. The actual vector U is horizontal written as entry 1 and 2, and the current approximation to the first eigenvalue is in the third column. The first(largest in absolute value) eigenvalue is negative, so the system needs a certain time to reduce oscillations:
in float reprecentation