Okay, I have a nice example please hold on while I post it.

Let's start with this simple book example and we will use a very simplified power method to get the dominant eigenpair. First we go after the dominant eigenvector using the power method.

We choose the initial vector with a guess of

We begin to iterate using x0 and A to generate x1.

Now for the purposes of keeping the elements of xn small we will divide x1 by the bottom entry in this case a 5. This was an arbitrary choice and in a real calculation we would use a better system.

Now we continue to iterate.

We divide x2 by the bottom element in this case 5.8

Continuing the above steps we get:

It really does look like the dominant eigenvector is approaching

From that eigenvector we get the dominant eigenvalue by applying this formula:

where x is the dominant eigenvector and A is original matrix ( above ).

That is the dominant eigenvalue. We are done.