# Numerical approximation of the eigenvalues and the eigenvector

1. Mar 12, 2012

### junsugal

1. The problem statement, all variables and given/known data

This problem will guide you through the steps to obtain a numerical approximation of the eigenvalues, and eigenvectors of A using an example.

We will define two sequences of vectors{vk} and {uk}
(a) Choose any vector u $\in$ R2 as u0
(b) Once uk has been determined, the vectors vk+1 and uk+1 are determined as follows:
i. Set vk+1 = Auk
ii. Find the entry of maximum absolute value of vk+1 lets say it is the j-th entry of vk+1
iii. Set uk+1 = vk+1/vjk+1
(c) The sequence {Uk} will converge to an eigenvector for A.

Let A =
2 1
1 2
Use the above method to approximate an eigenvector for A using only 4 iterations, that is finding v5.
Find the eigenvectors for A using the method learned in class and compare.

2. Relevant equations

3. The attempt at a solution

I assume u0 is (1,1)
From A, i found the eigenvalues which is 3 and 1.
And the corresponding eigenvectors is (1,1) and (-1,1)
Then i get uk as (3k-1, 3k+1)

Now, to the next step, I get my vk+1 as ( 2*(3k-1)+3k+1, 3k-1 + 2*(3k+1))
Plugging in k as 4 as I want to get v5
I get v5 as (242, 244)
And from this step the instruction said to find the entry of maximum absolute value of vk+1 and I don't know how. I am not even sure if what I've done is correct.

Please correct me if I am wrong and please explain on what to do next.