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I'm trying to understand dominant eigenvalues and I found this website that has a explanation of it (Definition 9.2). In the example, the power method is used to find the dominant eigenvector which correspondes to the eigenvalue of 1.

When I calculate the eigenvalues and vectors of the matrix in the example, I got this result:

enter image description here

The first line are the eigenvalues and the second row the eigenvectors. As you can see, the eigenvector that corresponde to to the eigenvalue of 1 is

{-0.577, -0.577, -0.577}

If I calculate the powers of the matrix, I find that after M^9, it converges as shown in the website

I don't understand what is the difference between the eigenvector that I found that corresponde to to the eigenvalue of 1 and the eigenvector that is found after elevating the matrix many times, and that the website described also as the eigenvector of eigenvalue 1.

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2 Answers 2

up vote 3 down vote accepted

What do you mean by "the eigenvector"? For each eigenvalue, there are infinitely many eigenvectors, and you typically get a basis as "the" eigenvectors.

From your post, it looks like the general eigenvector for $\lambda=1$ has the form $(t,t,t)$ and you obtained the one with $t= -0.57735$. But other values of $t$ lead to other eigenvectors, and if I remember right, the power method produces a probability eigenvector, thus in this case $t=\frac{1}{3}$.

P.S. Also note, if you are reffering to the first 3x3 example on that page (it would had been very helpfull if you included the matrix you calculated), you should note that they find the eigenvector for $**A^T**$. In that example, it is trivial to see that $(t,t,t)$ is an eigenvector for $A$, but they don't calculate the eigenvectors of $A$!

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Imagine someone has a list of numbers, known (to that person) but kept secret. Now imagine you have the ability to do some limited manipulation of those numbers and extract some information from the person about this set of numbers. You can ask that someone to alter every number in the same way by raising them to a power. Then you can ask that someone to assign each separate number to a specific (to that number) vector. The information then that you can get is the vector sum. If each but one number is smaller than unity in magnitude, they (the small numbers) will in the power exponential become small and their corresponding vector is scaled to a tiny bit. The number that is close or equal to one (or really just the largest in magnitude) will dominate, and the vector corresponding to that number will be apparent.

Now imagine all that again, but the person with the list of numbers is actually a matrix, the list of numbers are its eigenvalues, and the corresponding vectors for those numbers are the eigenvectors. That is what is called the power method.

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