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Math Help - 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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    M=\begin{bmatrix}<br />
0 & -2\\ <br />
4 & 6<br />
\end{bmatrix} and D^n=\begin{bmatrix}<br />
2^n & -2\\ <br />
0 & 8^n<br />
\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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    M^n=\begin{bmatrix}<br />
-1 & -1\\ <br />
2 & 1<br />
\end{bmatrix}\begin{bmatrix}<br />
4 & 0\\ <br />
0 & 2<br />
\end{bmatrix}^n\begin{bmatrix}<br />
-1 & -1\\ <br />
2 & 1<br />
\end{bmatrix}^{-1}
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  7. #7
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    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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