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Math Help - eigenspaces/diagonalizability

  1. #1
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    eigenspaces/diagonalizability

    Hey y'all I got a question regarding eigenspaces which is stumping me. Heres the problem, any tips or help would be greatly appreciated.

    Let T be an invertible linear operator on a finite-dimensional vector space V.
    Prove that the eigenspace of T corresponding to \lambda is the same as the eigenspace of T^{-1} corresponding to \lambda ^{-1}

    Also prove that if T is diagonalizable, then T^{-1} is diagonalizable.

    Thanks!
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  2. #2
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    Quote Originally Posted by dannyboycurtis View Post
    Hey y'all I got a question regarding eigenspaces which is stumping me. Heres the problem, any tips or help would be greatly appreciated.

    Let T be an invertible linear operator on a finite-dimensional vector space V.
    Prove that the eigenspace of T corresponding to \lambda is the same as the eigenspace of T^{-1} corresponding to \lambda ^{-1}

    Also prove that if T is diagonalizable, then T^{-1} is diagonalizable.

    Thanks!
    *********sigh********....really!

    Tv=\lambda v\Longrightarrow (apply\,\,T^{-1}\,\,on\,\,both\,\,sides)\,v=T{-1}(\lambda v) , and now just use that the inverse of a linear map is also linear.
    For diagonalizable: use the definition, apply, move sides...etc.

    Tonio
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