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Math Help - Eigenvalue and matrices

  1. #1
    Junior Member
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    Mar 2011
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    Eigenvalue and matrices

    Hi,
    I want to show that a matrix H that has an eigenvalue t can be used to show that the matrix \textbf{H}^n would have an eigenvalue of t^n.

    The formal way of doing this i would guess to be mathematical induction, though how would i do the inductive step? (As i couldn't take the determinant of the whole thing)

    thanks
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  2. #2
    MHF Contributor
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    Re: Eigenvalue and matrices

    If x is an eigenvector that has eigenvalue t, then Hx=tx. Therefore, H^{n+1}x=H(H^nx)=H(t^nx)=t^n(Hx)=t^{n+1}x.
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