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Math Help - [SOLVED] eigenvalue help!

  1. #1
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    [SOLVED] eigenvalue help!

    If a is an eigenvalue of A. show that a^(-1) is an eigenvalue of A^-1. I don't know where to start.
    Last edited by Jhevon; March 30th 2009 at 09:13 PM. Reason: Restored problem
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  2. #2
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    Quote Originally Posted by lord12 View Post
    If x is an eigenvalue of A. show that a^(-1) is an eigenvalue of A^-1. I don't know where to start.
    Do you mean that if a is an eigenvalue of A, show that a^{-1} is an eigenvalue of A^{-1}?
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    yes!
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    Quote Originally Posted by lord12 View Post
    yes!
    If a is an eigenvalue of A, then

    A\mathbf{x} = a\mathbf{x} for some eigenvector \mathbf{x}.

    Notice that a\mathbf{x} = aI\mathbf{x}, where I is the Identity matrix.

    Therefore

    A\mathbf{x} - aI\mathbf{x} = \mathbf{0}

    (A - aI)\mathbf{x} = \mathbf{0}.

    For a nonzero \mathbf{x}, A - aI = \mathbf{0}.


    Therefore A = aI.

    If A is nonsingular, then

    A^{-1} = (aI)^{-1}.


    By definition of inverses

    AA^{-1} = aI(aI)^{-1} = I

    aI(aI)^{-1} = I

    I(aI)^{-1} = a^{-1}I

    (aI)^{-1} = a^{-1}I

    A^{-1} = a^{-1}I.


    Therefore a^{-1} is an eigenvalue of A^{-1}.
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