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Inverse Power Method and Eigenvectors

  1. Dec 2, 2011 #1
    1. The problem statement, all variables and given/known data

    The Markov matrix A = [.9 .3; .1 .7] has eigenvalues 1 and .6, and the power method uk=Aku0 converges to [.75 .25]T. Find the eigenvectors of A-1. What does the inverse power method u-k=A-1u0 converge to (after you multiply by .6k)?

    2. Relevant equations



    3. The attempt at a solution

    Eigenvalue 1 is the dominant one when using the power method on A. However, we're interested in the smallest eigenvalue when dealing with the inverse power method, in this case .6. The eigenvalues of A-1 are:
    (1/.6) and 1. According to theory, the eigenvectors of A-1 are the same as those in A.

    So, the corresponding eigenvector to the value .6 is [-1 1]T.

    From there, I'm simply stumped. Can anyone please help?!
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Dec 2, 2011 #2
    Also, I found that the eigenvector corresponding to the eigenvalue 1 is [3 1]T. Still confused though... Not sure how to proceed.
     
  4. Dec 3, 2011 #3
    I know that these eigenvectors correspond to the eigenvalues of A-1, and these eigenvalues are the reciprocal of those given. Does anyone know how to apply the power method to A-1? Any ideas? Thanks!
     
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