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Numerical approximation of the eigenvalues and the eigenvector

  1. Mar 12, 2012 #1
    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 [itex]\in[/itex] 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.
    Thanks in advance!
     
  2. jcsd
  3. Mar 12, 2012 #2
    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
     
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