Numerical approximation of the eigenvalues and the eigenvector

[junsugal]

Homework Statement

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{v_k} and {u_k}
(a) Choose any vector u \in R² as u₀
(b) Once u_k has been determined, the vectors v_k+1 and u_k+1 are determined as follows:
i. Set v_k+1 = Au_k
ii. Find the entry of maximum absolute value of v_k+1 let's say it is the j-th entry of v_k+1
iii. Set u_k+1 = v_k+1/v_jk+1
(c) The sequence {U_k} 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.

Homework Equations

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 u_k as (3^k-1, 3^k+1)

Now, to the next step, I get my v_k+1 as ( 2*(3^k-1)+3^k+1, 3^k-1 + 2*(3^k+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 v_k+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.
Thanks in advance!

 

[sunjin09]

You chose u_0=(1,1), you would get u_k=(3^k,3^k). Even if you got v_5=(242,244) somehow, the max abs entry is obviously 244, because |244|>|242|. So u_5=(242/244,1).
Now find another less clever and more random u_0 and try again