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Math Help - Limits of matrices

  1. #1
    Junior Member pearlyc's Avatar
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    Limits of matrices

    Calculate lim n->infinity A^n for large n where

    \left[ \begin{array}{cccc} 1/2 & 0 & 0 \\ 1/2 & 1 & 3/4 \\ 0 & 0 & 1/4\end{array}\right]
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  2. #2
    Moo
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    Hello,

    Quote Originally Posted by pearlyc View Post
    Calculate lim n->infinity A^n for large n where

    \left[ \begin{array}{cccc} 1/2 & 0 & 0 \\ 1/2 & 1 & 3/4 \\ 0 & 0 & 1/4\end{array}\right]
    Show by induction that :

    A^n=\begin{pmatrix} \frac{1}{2^n} & 0 & 0 \\ (1-\frac{1}{2^n}) & 1 & (1-\frac{1}{4^n}) \\ 0 & 0 & \frac{1}{4^n} \end{pmatrix}

    Then, find its limit
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  3. #3
    Super Member flyingsquirrel's Avatar
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    Hi

    Another solution has been given in this thread.
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    Junior Member pearlyc's Avatar
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    omg, how do you do that ><?
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  5. #5
    Lord of certain Rings
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    The alternate but more mechanical way of doing this is:

    Diagonalize the matrix: That is find an invertible P such that P^{-1}AP = D. From this we see that P^{-1}A^n P = D^n.

    Here the eigenvalues are 1,\frac12, \frac14

    With P = \begin{pmatrix} 0 & 1 & 0\\ 1 & -1 & 0 \\ 0 & -1 & 3 \end{pmatrix}

    P^{-1}AP = D = \begin{pmatrix} 1 & 0 & 0\\ 0 & \frac12 & 0 \\ 0 & 0 & \frac14 \end{pmatrix}

    P^{-1}A^n P = D^n = \begin{pmatrix} 1 & 0 & 0\\ 0 & \left(\frac12\right)^n & 0 \\ 0 & 0 & \left(\frac14\right)^n \end{pmatrix}

    So
    \lim_{n \to \infty}P^{-1}A^n P = \lim_{n \to \infty}  \begin{pmatrix} 1 & 0 & 0\\ 0 & \left(\frac12\right)^n & 0 \\ 0 & 0 & \left(\frac14\right)^n \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0\\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}

    \lim_{n \to \infty}A^n  = P \begin{pmatrix} 1 & 0 & 0\\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} P^{-1}
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