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Diagonalizable Matrix

  1. Mar 17, 2008 #1
    1. The problem statement, all variables and given/known data
    Let A =
    [4 0 1;
    2 3 2;
    1 0 4]
    Let n >= 1 be an integer. Compute the matrix A^n with entries depending on n.

    2. Relevant equations

    3. The attempt at a solution
    First I need to show that A is diagonalizable, and find a matrix S such that D = (S^-1)(A)(S) is diagonal. I am having serious trouble finding S. First I found the eigenvalues of A to be 3,3, and 5. Next I found the eigenvectors to be for [tex]\lambda = 3[/tex]: (1 0 -1) and [tex]\lambda = 5[/tex] : (1 2 1). There's a theorem in my book that says that a matrix is only diagonalizable if and only if there are n linearly independent eigenvectors. I only have 2. I'm not quite sure how to get the third.
    Now, assuming I had three eigenvectors, I would combine them to form the matrix S, and if the [tex]det(S)\neq 0[/tex] then the eigenvectors would be linearly independent, and
    i could find S^-1. Then A^n would just be (S^-1)(D^n)(S). Does this all sound correct? I guess I'm just a little confused about the eigenvalue/eigenvector situation....
    Thanks for any help.
     
  2. jcsd
  3. Mar 17, 2008 #2

    Dick

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    Science Advisor
    Homework Helper

    There is a third linearly independent eigenvector, you are just overlooking it. Hint: try (0,1,0). Then do exactly what you propose.
     
  4. Mar 17, 2008 #3
    Wouldn't An = SDnS-1?
     
  5. Mar 17, 2008 #4

    Dick

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    Science Advisor
    Homework Helper

    Yep. Sorry, I overlooked that.
     
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