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Math Help - Eigenvectors

  1. #1
    Senior Member I-Think's Avatar
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    Eigenvectors

    The matrix A=\left(\begin{array}{ccc}3&-1&0\\-4&-6&-6\\5&11&10\end{array}\right)

    has eigenvectors \left(\begin{array}{c}1\\-1\\1\end{array}\right), \left(\begin{array}{c}1\\2\\-3\end{array}\right) and \left(\begin{array}{c}1\\1\\-2\end{array}\right) with corresponding eigenvalues 4, 1 and 2 respectively.
    The matrix B has eigenvalues 2, 3, 1 with corresponding eigenvectors
    \left(\begin{array}{c}1\\-1\\1\end{array}\right), \left(\begin{array}{c}1\\2\\-3\end{array}\right) and \left(\begin{array}{c}1\\1\\-2\end{array}\right)
    Find a matrix P and a diagonal matrix D such that
    (A+B)^4=PDP^{-1}

    Not required to evaluate P^{-1}

    A start and some hints would be very nice.
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  2. #2
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    Quote Originally Posted by I-Think View Post
    The matrix A=\left(\begin{array}{ccc}3&-1&0\\-4&-6&-6\\5&11&10\end{array}\right)

    has eigenvectors \left(\begin{array}{c}1\\-1\\1\end{array}\right), \left(\begin{array}{c}1\\2\\-3\end{array}\right) and \left(\begin{array}{c}1\\1\\-2\end{array}\right) with corresponding eigenvalues 4, 1 and 2 respectively.
    The matrix B has eigenvalues 2, 3, 1 with corresponding eigenvectors
    \left(\begin{array}{c}1\\-1\\1\end{array}\right), \left(\begin{array}{c}1\\2\\-3\end{array}\right) and \left(\begin{array}{c}1\\1\\-2\end{array}\right)
    Find a matrix P and a diagonal matrix D such that
    (A+B)^4=PDP^{-1}

    Not required to evaluate P^{-1}

    A start and some hints would be very nice.
    Notice that the eigenvectors for B are the same as those of A. What do you know about the matrix whose columns are those three eigenvectors?
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  3. #3
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    And of course, if v is an eigenvector of A with eigenvalue \lambda_A and an eigenvector of B with eigenvalue \lambda_B then (A+ B)v= Av+ Bv= \lambda_A v+ \lambda_B v= (\lambda_A+ \lambda_B) v so v is an eigenvector for A+ B with eigenvalue \lambda_A+ \lambda_B. That gives you the diagonal matrix D.
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  4. #4
    Senior Member I-Think's Avatar
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    Thanks Opalq, I actually realize that they had the same eigenvectors.
    And HallsofIvy, that was actually the first part of the question.

    Just one last question, for the matirx P, does it matter what order the eigenvectors are placed ?
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  5. #5
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    Quote Originally Posted by I-Think View Post
    Just one last question, for the matrix P, does it matter what order the eigenvectors are placed ?
    No, but the eigenvalues on the diagonal of D must occur in the same order as their corresponding eigenvectors in the columns of P.
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