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Math Help - Eigenvalues and Eigenvectors raised to a power.

  1. #1
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    Eigenvalues and Eigenvectors raised to a power.

    If you have a matrix, and raise it to the power n, what would be the effect on the eigenvalues and the eigenvectors?

    Thanks
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  2. #2
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    Quote Originally Posted by llamagoogle View Post
    If you have a matrix, and raise it to the power n, what would be the effect on the eigenvalues and the eigenvectors?

    Thanks

    Av=\lambda v\Longrightarrow A^nv=\lambda^nv\,,\,\,\forall\,n\in\mathbb{N} . You can easily prove this by induction.

    If the matrix is invertible the above is extendable to negative powers, too.

    Tonio
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  3. #3
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    Thank you,
    Where can I find how to prove this?
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    Quote Originally Posted by llamagoogle View Post
    Thank you,

    Where can I find how to prove this?

    In your head (honest: the proof is very easy), but I guess books in algebra or linear algebra must deal with this, though I think there's a good chance it'll be an exercise because, as already said, it is very easy. Have you REALLY tried to prove it by yourself already?

    Tonio
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