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Math Help - Cayley Hamilton Theorem and raising a 2x2 matrix to the nth power

  1. #1
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    Cayley Hamilton Theorem and raising a 2x2 matrix to the nth power

    A 2x2 matrix has (in this case) 2 distinct eigenvalues. The matrix to the power n can be expressed as a coefficient times the matrix plus a second coefficient times the identity matrix. Can someone point me to a simple derivation of a formula that computes the two coefficients? That is, find the two coefficients in terms of the two eigenvalues and n. Yes, I know the problem can also be soved by diagonalizing the matrix and then raising the diagonal elements to the power n, but I want to avoid computing and then multiplying by the pre and post-matrices that are required for that method. There are (at least) two ways to skin this cat, and I'm asking for help on the method that does not involve explicit diagonalization. Many thanks.
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  2. #2
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    Re: Cayley Hamilton Theorem and raising a 2x2 matrix to the nth power

    I think in this case, Cayley Hamilton got in your way, instead of helping. This is what you want, I believe:

    Cayley Hamilton Theorem and raising a 2x2 matrix to the nth power-mhfsimplematrix.png
    Thanks from joemarasco
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  3. #3
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    Re: Cayley Hamilton Theorem and raising a 2x2 matrix to the nth power

    Thank you very much. Your solution is very simple, elegant, and easy to understand. Actually, I used CH to figure out that an arbitrary 2x2 matrix with distinct eigenvalues raised to the nth power could be expressed as a linear combination of itself and the 2x2 identity matrix, with appropriate coefficients. Your solution ties everything together, by starting from the diagonal case and then reasoning that my original matrix could be put into that form. You got to the formula I was searching without having to compute the transformation matrix and its inverse explicitly, which was the objective of the original question. Nicely done, and once again, many thanks.
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