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Math Help - Matrix Eigenvectors

  1. #1
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    Matrix Eigenvectors

    A is a 3 x 3 matrix with eigenvectors v1 = [1 0 0], v2 = [1 1 0], v3 = [1 1 1] (all vertical) corresponding to eigenvalues x1 = -1/3, x2 = 1/3, x3 = 1, respectively and x = [2 1 2].

    Find (A^k)x. What happens as k approaches infinity?

    I don't even know where to begin on this one.

    Thanks for the help!
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  2. #2
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    First use the eigenvalue equation (Av = lambda*v) to backsolve for the values in A. You'll also have to use the property that the trace of a matrix (sum of the diagonal values) is also equal to the sum of that matrix's eigenvalues.

    Once you have A, you'll see it is an upper-triangular matrix with values <= 1....and you can see what will happen if you raise this matrix to a high power. (if you can't see it, just type it into matlab and do A^5, A^6, ... until you see what is happening)
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  3. #3
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    Quote Originally Posted by Laydieofsorrows View Post
    A is a 3 x 3 matrix with eigenvectors v1 = [1 0 0], v2 = [1 1 0], v3 = [1 1 1] (all vertical) corresponding to eigenvalues x1 = -1/3, x2 = 1/3, x3 = 1, respectively and x = [2 1 2].

    Find (A^k)x. What happens as k approaches infinity?

    I don't even know where to begin on this one.

    Thanks for the help!
    First, note that x=2v_3-v_2+v_1. Now,

    A^kx = A^k(2v_3-v_2+v_1) = 2(A^kv_3) - A^kv_2 + A^kv_1.

    Use the fact that if \lambda is an eigenvalue of a matrix C with eigenvector v then \lambda ^k is en eigenvalue of C^k with eigenvector v (prove this if you haven't already) to get:

    A^kx = 2(x_3^kv_3) - x_2^kv_2 + x_1^kv_1

    and finish.
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