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Matrix A and its inverse have the same eigenvectors

  1. Dec 7, 2016 #1
    1. The problem statement, all variables and given/known data
    T/F: Each eigenvector of an invertible matrix A is also an eignevector of A-1

    2. Relevant equations


    3. The attempt at a solution
    I know that if A is invertible and ##A\vec{v} = \lambda \vec{v}##, then ##A^{-1} \vec{v} = \frac{1}{\lambda} \vec{v}##, which seems to imply that A and its inverse have the same eigenvectors. However, what about if A has all distinct eigenvalues, then ##A = PDP^{-1}##. From this, we conclude that ##A^{-1} = P^{-1} D^{-1} P##. Doesn't this show that A and its inverse have different eigenvectors? Since for A the matrix of eigenvectors is ##P## while for A-1 the matrix of eigenvectors is ##P^{-1}##?
     
  2. jcsd
  3. Dec 7, 2016 #2

    fresh_42

    Staff: Mentor

    It's ##A^{-1}=(PDP^{-1})^{-1}=PD^{-1}P^{-1}##. You have provided the argument that is needed, and you showed, that the eigenvalues are inverse, too, e.g. the elements in ##D## and ##D^{-1}##. Why should there be a discrepancy?
     
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