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Thread: Diagonalization

  1. #1
    Junior Member krtica's Avatar
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    Diagonalization

    Let 0-2 46 .
    q: Find formulas for the entries of , where is a positive integer.

    I found eigenvalues, 2 and 4, then found their corresponding eigenspaces.

    D^n=(P^-1)(A^n)P
    =
    2^n-2 0
    8^n
    .
    Which is incorrect. Can someone please help?
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  2. #2
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    Quote Originally Posted by krtica View Post
    Let 0-2 46 .
    q: Find formulas for the entries of , where is a positive integer.

    I found eigenvalues, 2 and 4, then found their corresponding eigenspaces.

    D^n=(P^-1)(A^n)P
    =
    2^n-2 0
    8^n
    .
    Which is incorrect. Can someone please help?
    What does this notation mean?
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  3. #3
    Junior Member krtica's Avatar
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    I apologize, the format didn't turn out as I expected. It is a 2x2 matrix. The first row is 2^n, -2. The second row is 0, 8^n.

    D=(P^-1)AP, where P is a basis for both Eigenspaces with eigenvalues of 2 and 4. A is for the original matrix.
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  4. #4
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    $\displaystyle M=\begin{bmatrix}
    0 & -2\\
    4 & 6
    \end{bmatrix}$ and $\displaystyle D^n=\begin{bmatrix}
    2^n & -2\\
    0 & 8^n
    \end{bmatrix}$ correct?
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  5. #5
    Junior Member krtica's Avatar
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    Yes, that's correct.
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  6. #6
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    $\displaystyle M^n=\begin{bmatrix}
    -1 & -1\\
    2 & 1
    \end{bmatrix}\begin{bmatrix}
    4 & 0\\
    0 & 2
    \end{bmatrix}^n\begin{bmatrix}
    -1 & -1\\
    2 & 1
    \end{bmatrix}^{-1}$
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  7. #7
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    $\displaystyle M^n=XD^nX^{-1}$ where D is the a diagonal matrix of the eigenvalues and X is the matrix with the corresponding eigenvectors.
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  8. #8
    Junior Member krtica's Avatar
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    Thank you. I really, really appreciate your help.
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